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Fourth Edition 

J.W. Mullin 

Emeritus Professor of Chemical Engineering, 
University of London 

E I N E M A N N 


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Linacre House, Jordan Hill, Oxford 0X2 8DP 

225 Wildwood Avenue, Woburn, MA 01801-2041 

A division of Reed Educational and Professional Publishing Ltd 

-^J A member of the Reed Elsevier pic group 

First published 1961 
Second edition 1972 
Third edition 1992 
Reprinted 1994, 1995 
Paperback edition 1997 
Fourth edition 2001 

© Reed Educational and Professional Publishing Ltd 2001 

All rights reserved. No part of this publication 
may be reproduced in any material form (including 
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means and whether or not transiently or incidentally 
to some other use of this publication) without the 
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Designs and Patents Act 1988 or under the terms of a 
licence issued by the Copyright Licensing Agency Ltd, 
90 Tottenham Court Road, London, England W1P 9HE. 
Applications for the copyright holder's written permission 
to reproduce any part of this publication should be addressed 
to the publishers 

British Library Cataloguing in Publication Data 

A Catalogue record for this book is available from the British Library 

Library of Congress Cataloguing in Publication Data 

A Catalogue record for this book is available from the Library of Congress 

ISBN 7506 4833 3 

Typeset in India at Integra Software Services Pvt Ltd, Pondicherry, 
India 605005; 

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Preface to Fourth Edition viii 

Preface to First Edition x 

Nomenclature and units xii 

1 The crystalline state 1 

1.1 Liquid crystals 1 

1.2 Crystalline solids 3 

1.3 Crystal symmetry 4 

1.4 Crystal systems 7 

1.5 Miller indices 10 

1.6 Space lattices 13 

1.7 Solid state bonding 15 

1.8 Isomorphs and polymorphs 16 

1.9 Enantiomorphs and chirality 18 

1.10 Crystal habit 22 

1.11 Dendrites 24 

1.12 Composite crystals and twins 25 

1.13 Imperfections in crystals 27 

2 Physical and thermal properties 32 

2.1 Density 32 

2.2 Viscosity 35 

2.3 Surface tension 39 

2.4 Diffusivity 41 

2.5 Refractive index 47 

2.6 Electrolytic conductivity 48 

2.7 Crystal hardness 48 

2.8 Units of heat 49 

2.9 Heat capacity 50 

2.10 Thermal conductivity 54 

2.11 Boiling, freezing and melting points 55 

2.12 Enthalpies of phase change 58 

2.13 Heats of solution and crystallization 62 

2.14 Size classification of crystals 64 

3 Solutions and solubility 86 

3.1 Solutions and melts 86 

3.2 Solvent selection 86 

3.3 Expression of solution composition 90 

3.4 Solubility correlations 92 

3.5 Theoretical crystal yield 96 

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3.6 Ideal and non-ideal solutions 


h 7 Particle size and snlnhility I 

Effect of impurities on solubilitvl 
Measurement of solubility! 

3.10 Prediction of solubility 

3.11 Solubility data sources 

3.12 Supersolubility 

|3.13 Solution structure | 



Phase equilibria 

The phase rule 

4.2 One-component systems I 

1.3 Two-component systems I 

4.4 Enthalpy-composition diagrams I 

1 4.5 Phase change detection | 

14.6 Three-component svstemsl 
|4.7 Four-component systems! 
14.8 'Dynamic' phase diagrams! 

1 5 NucleatiorTI 

|5.1 Primary nucleationl 

15.2 Secondary nuclcationl 

15.3 Metastable zone widths I 

1 5.4 Effect of impurities! 

|5.5 Induction and latent periods"! 

1 5.6 Interfacial tension | 

I 5.7 Ostwald's rule of stages | 




6 Crystal growth 

6.1 Crystal growth theories 

6.2 Growth rate measurements I 

16.3 Crystal growth and dissolution | 

I 6.4 Crystal habit modification I 

1 6.5 Polymorphs and phase transformations - ! 

16.6 Inclusions 1 


7 Recrystallization 

7.1 Recrystallization schemes 

I 7.2 Resolution of racemates| 



7.3 Isolation of polymorphs I 

Recrystallization from supercritical fluidt 
Zone refining | 
Single crystals | 


S Industrial techniques and equipment 

. 1 Precipitation 

8.2 Crystallization from melts 



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Contents vii 

8.3 Sublimation 358 

8.4 Crystallization from solution 368 

9 Crystallizer design and operation 403 

9.1 Crystal size distribution (CSD) 403 

9.2 Kinetic data measurement and utilization 430 

9.3 Crystallizer specification 434 

9.4 Fluid-particle suspensions 451 

9.5 Encrustation 459 

9.6 Caking of crystals 463 

9.7 Downstream processes 467 

Appendix 478 

References 536 

Author index 577 

Subject index 587 

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Preface to Fourth Edition 

This fourth edition of Crystallization has been substantially rewritten and 
up-dated. The 1961 first edition, written primarily for chemical engineers and 
industrial chemists, was illustrated with practical examples from a range of 
process industries, coupled with basic introductions to the scientific principles 
on which the unit operation of crystallization depends. It was also intended to 
be useful to students of chemical engineering and chemical technology. The 
aims and objectives of the book have remained intact in all subsequent editions, 
although the subject matter has been considerably expanded each time to take 
into account technological developments and to reflect current research trends 
into the fundamentals of crystallization mechanisms. 

The continuing upsurge in interest in the utilization of crystallization as a 
processing technique covers an increasing variety of industrial applications, not 
only in the long-established fields of bulk inorganic and organic chemical 
production, but also in the rapidly expanding areas of fine and specialty 
chemicals and pharmaceuticals. These developments have created an enormous 
publication explosion over the past few decades, in a very wide range of 
journals, and justify the large number of specialist symposia that continue to 
be held world-wide on the subject of crystallization. 

Particular attention is drawn in this edition to such topical subjects as 
the isolation of polymorphs and resolution of enantiomeric systems, the 
potential for crystallizing from supercritical fluids, the use of molecular 
modelling in the search for tailored habit modifiers and the mechanisms of 
the effect of added impurities on the crystal growth process, the use of com- 
puter-aided fluid dynamic modelling as a means of achieving a better under- 
standing of mixing processes, the separate and distinct roles of both batch 
and continuous crystallization processing, and the importance of potential 
downstream processing problems and methods for their identification from 
laboratory investigations. Great care has been taken in selecting suitable liter- 
ature references for the individual sections to give a reliable guide to further 

Once again I want to record my indebtedness to past research students, 
visiting researchers and colleagues in the Crystallization Group at University 
College London over many years, for their help and support in so many ways. 
They are too numerous to name individually here, but much of their work is 
recorded and duly acknowledged in appropriate sections throughout this edition. 
I should like to express my sincere personal thanks to them all. I am also very 
grateful to all those who have spoken or written to me over the years with 
useful suggestions for corrections or improvements to the text. 

Finally, and most importantly, it gives me great pleasure to acknowledge the 
debt I owe to my wife, Averil, who has assisted me with all four editions of 

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Preface to Fourth Edition ix 

Crystallization. Without her tremendous help in preparing the manuscripts, my 
task of writing would not have been completed. 

University College London 

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Preface to First Edition 

Crystallization must surely rank as the oldest unit operation, in the chemical 
engineering sense. Sodium chloride, for example, has been manufactured by 
this process since the dawn of civilization. Today there are few sections of the 
chemical industry that do not, at some stage, utilize crystallization as a method 
of production, purification or recovery of solid material. Apart from being one 
of the best and cheapest methods available for the production of pure solids 
from impure solutions, crystallization has the additional advantage of giving an 
end product that has many desirable properties. Uniform crystals have good 
flow, handling and packaging characteristics: they also have an attractive 
appearance, and this latter property alone can be a very important sales factor. 

The industrial applications of crystallization are not necessarily confined to 
the production of pure solid substances. In recent years large-scale purification 
techniques have been developed for substances that are normally liquid at room 
temperature. The petroleum industry, for example, in which distillation has 
long held pride of place as the major processing operation, is turning its 
attention most keenly to low-temperature crystallization as a method for the 
separation of 'difficult' liquid hydrocarbon mixtures. 

It is rather surprising that few books, indeed none in the English language, 
have been devoted to a general treatment of crystallization practice, in view of 
its importance and extensive industrial application. One reason for this lack of 
attention could easily be that crystallization is still referred to as more of an art 
than a science. There is undoubtedly some truth in this old adage, as anyone 
who has designed and subsequently operated a crystallizer will know, but it 
cannot be denied that nowadays there is a considerable amount of science 
associated with the art. 

Despite the large number of advances that have been made in recent years in 
crystallization technology, there is still plenty of evidence of the reluctance to 
talk about crystallization as a process divorced from considerations of the 
actual substance being crystallized. To some extent this state of affairs is similar 
to that which existed in the field of distillation some decades ago when little 
attempt had been made to correlate the highly specialized techniques devel- 
oped, more or less independently, for the processing of such commodities as 
coal tar, alcohol and petroleum products. The transformation from an 'art' to a 
'science' was eventually made when it came to be recognized that the key factor 
which unified distillation design methods lay in the equilibrium physical prop- 
erties of the working systems. 

There is a growing trend today towards a unified approach to crystallization 
problems, but there is still some way to go before crystallization ceases to be the 
Cinderella of the unit operations. More data, particularly of the applied kind, 
should be published. In this age of prolific outputs of technical literature such 
a recommendation is not made lightly, but there is a real deficiency of this type 

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Preface to First Edition xi 

of published information. There is, at the same time, a wealth of knowledge and 
experience retained in the process industries, much of it empirical but none the 
less valuable when collected and correlated. 

The object of this book is to outline the more important aspects of crystal- 
lization theory and practice, together with some closely allied topics. The book 
is intended to serve process chemists and engineers, and it should prove of 
interest to students of chemical engineering and chemical technology. While 
many of the techniques and operations have been described with reference to 
specific processes or industries, an attempt has been made to treat the subject 
matter in as general a manner as possible in order to emphasize the unit 
operational nature of crystallization. Particular attention has been paid to the 
newer and more recently developed processing methods, even where these have 
not as yet proved adaptable to the large-scale manufacture of crystals. 

My thanks are due to the Editors of Chemical Engineering Practice for 
permission to include some of the material and many of the diagrams pre- 
viously published by me in Volume 6 of their 12-volume series. I am indebted to 
Professor M. B. Donald, who first suggested that I should write on this subject, 
and to many of my colleagues, past and present, for helpful discussions in 
connection with this work. I would also like to take this opportunity of 
acknowledging my indebtedness to my wife for the valuable assistance and 
encouragement she gave me during the preparation of the manuscript. 




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Nomenclature and units 

The basic SI units of mass, length and time are the kilogram (kg), metre (m) and 
second (s). The basic unit of thermodynamic temperature is the kelvin (K), but 
temperatures and temperature differences may also be expressed in degrees 
Celsius (°C). The unit for the amount of substance is the mole (mol), defined 
as the amount of substance which contains as many elementary units as there 
are atoms in 0.012 kg of carbon- 12. Chemical engineers, however, are tending 
to use the kilomole (kmol = 10 3 mol) as the preferred unit. The unit of electric 
current is the ampere (A). 

Several of the derived SI units have special names: 




SI unit 

Basic SI unit 




s- 1 




mkgs~ 2 




Nm~ 2 

m _1 kgs~ 2 

Energy, work; heat 




m 2 kg s~ 2 




Js- 1 

m 2 kg s~ 3 

Quantity of electricity 




Electric potential 




m 2 kgs~ 3 A -1 

Electric resistance 




m 2 kgs~ 3 A~ 2 





m _2 kg _1 s 3 A 2 




cv- 1 

m- 2 kg-'s 4 A 2 

Magnetic flux 




m 2 kgs~ 2 A -1 

Magnetic flux density 



Wbm- 2 

kgs- 2 A-' 




\VbA~ 1 

m 2 kg s~ 2 A~ 2 

Up to the present moment, there is no general acceptance of the pascal for 
expressing pressures in the chemical industry; many workers prefer to use 
multiples and submultiples of the bar (1 bar = 10 5 Pa = 10 5 Nm~ 2 « 1 atmos- 
phere). The standard atmosphere (760mmHg) is defined as 1.0133 x 10 5 Pa, 
i.e. 1.0133 bar. 

The prefixes for unit multiples and submultiples are: 

10~ 18 atto a 

10~ 15 femto f 

10~ 12 pico p 

10~ 9 nano n 

10~ 6 micro \i 

10~ 3 milli m 

10~ 2 centi c 

10-' deci d 




10 2 



10 3 



10 6 



10 9 



10 12 



10 15 



10 18 



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Nomenclature and units 

Conversion factors for some common units used in chemical engineering are 
listed below. An asterisk (*) denotes an exact relationship. 







Temperature difference 
Energy (work, heat) 

1 in 

25.4 mm 


0.3048 m 



1 mile 

1.6093 km 

1 A (angstrom) 

10- 10 m 

1 min 



3.6 ks 

1 day 

86.4 ks 

1 year 

31.5 Ms 

1 • 2 

1 in 

645.16 mm 2 

1ft 2 

0.092903 m 2 

1yd 2 

0.83613m 2 

1 acre 

4046.9 m 2 

1 hectare 

10 000 m 2 

1 mile 2 

2.590 km 2 

1 • 3 

1 in 

16.387 cm 3 

1ft 3 

0.02832 m 3 

1yd 3 

0.76453 m 3 

1 UK gal 

4546.1cm 3 

1 US gal 

3785.4 cm 3 

1 oz 


1 grain 

0.06480 g 


045359237 k 

1 cwt 

508023 kg 

1 ton 

1016.06 kg 




4.4482 N 


9.8067 N 


9.9640 kN 

1 dyn 

10~ 5 N 

1 degF (degR) 






1 cal (internat. table) 



10 7 J 


1.05506 kJ 

1 chu 

1.8991 kJ 


2.6845 MJ 


3.6 MJ 

1 therm 

105.51 MJ 

1 thermie 


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xiv Nomenclature and units 
Calorific value (volumetric) 


Volumetric flow 

Mass flow 

Mass per unit area 



Power (heat flow) 

Btu/ft 3 

chu/ft 3 

kcal/ft 3 

kcal/m 3 

therm/ft 3 


ft 3 /s 
ft 3 /h 




lb/in 2 
lb/ft 2 
ton/mile 2 

lb/in 3 
lb/ft 3 
lb/UK gal 
lb/US gal 

lbf/in 2 

tonf/in 2 

lbf/ft 2 

kgf/m 2 

standard atm 

at (1 kgf/cm 2 ) 


ft water 

in water 


mmHg (1 torr) 

hp (British) 

hp (metric) 


ft lbf/s 






ton of refrigeration 

37.259 kJrrr 3 
67.067 kJm- 3 
147.86 kJm- 3 
4.1868 kJm- 3 
3.7260 GJrrr 3 

0.3048 ms- 1 
5.0800 mm s" 1 
84.66711ms" 1 
0.44704 ms- 1 

0.028316m 3 s- 1 
7.8658 cm 3 s- 1 
1.2628 cm 3 s- 1 
1.0515cm 3 s" 1 

0. 12600 gs- 1 
0.28224 kg s- 1 

703.07 kg m- 2 
4.8824 kg m- 2 
392.30 kg km" 2 

27.680 gem" 3 
16.019kgm- 3 
99.776 kg m- 3 
119.83kgm- 3 

6.8948 kNm- 2 
1 5.444 MNm- 2 
47.880 Nm- 2 
9.8067 Nm- 2 
101.325 kNm" 2 
98.0665 kNm- 2 
10 5 Nrrr 2 
2.9891 kNm" 2 
249.09 Nm" 2 
3.3864 kNm" 2 
133.32Nm- 2 

745.70 W 
735. 50W 

io- 7 w 

0.29308 W 
1.0551 kW 
1.8991 kW 

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Nomenclature and units 

Moment of inertia 

lib ft 2 

0.042140 kg m 2 



0.13826 kgms- 1 

Angular momentum 

1 lb ft 2 /s 

0.042 140 kg m 2 s-! 

Viscosity, dynamic 

*1 poise (1 g/cms) 

0.1Nsm~ 2 
(O.lkgrrr's- 1 ) 

1 lb/ft h 

0.41338 mNsirr 2 

1 lb/ft s 

1.4882 NsirT 2 

Viscosity, kinematic 

*1 stokes (1 cm 2 /s) 

10- 4 m 2 s- 1 

1 ft 2 /h 

0.25806 cm 2 s-' 

Surface energy 

1 erg/cm 2 

10- 3 Jm- 2 

(surface tension) 

(1 dyn/cm) 

(lO^Nm" 1 ) 

Surface per unit volume 

1 ft 2 /ft 3 

3.2808 m 2 m- 3 

Surface per unit mass 

1 ft 2 /lb 

0.20482 m 2 kg" 1 

Mass flux density 

llb/hft 2 

1.3562gs-'m- 2 

Heat flux density 

1 Btu/h ft 2 

3.1546Wm- 2 

*lkcal/hm 2 

1.163Wrrr 2 

Heat transfer 

1 Btu/h ft 2o F 

5.6784 Wm- 2 K-' 


lkcal/hm 2o C 

F^OWni^R- 1 

Specific enthalpy 

* 1 Btu/lb 

2.326 kJ kg" 1 

(latent heat, etc.) 

Heat capacity 


4.1868 kJ kg-' K- 1 

(specific heat) 

Thermal conductivity 

1 Btu/h ft°F 

1.7307Wm- 1 K- 1 


l.^SWrn-'K- 1 

The values of some common physical constants in SI units include: 

Avogadro number, Na 6.023 x 10 23 mol _1 

Boltzmann constant, k 1.3805 x 10~ 23 JK~' 

Planck constant, h 6.626 x 10~ 34 Js 

Stefan-Boltzmann constant, a 5.6697 x 10" 8 Wm" 2 K" 4 

Standard temperature and pressure 273.15 K and 1.013 x 10 5 Nm- 2 


Volume of 1 kmol of ideal gas at s.t.p. 22.41 m 3 

Gravitational acceleration 9.807 ms- 2 

Universal gas constant, R 8.3143 J mol-' K _l 

Faraday constant, F 9.6487 x lO^moF 1 

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1 The crystalline state 

The three general states of matter - gaseous, liquid and solid - represent very 
different degrees of atomic or molecular mobility. In the gaseous state, the 
molecules are in constant, vigorous and random motion; a mass of gas takes 
the shape of its container, is readily compressed and exhibits a low viscosity. In 
the liquid state, random molecular motion is much more restricted. The volume 
occupied by a liquid is limited; a liquid only takes the shape of the occupied 
part of its container, and its free surface is flat, except in those regions where it 
comes into contact with the container walls. A liquid exhibits a much higher 
viscosity than a gas and is less easily compressed. In the solid state, molecular 
motion is confined to an oscillation about a fixed position, and the rigid 
structure generally resists compression very strongly; in fact it will often frac- 
ture when subjected to a deforming force. 

Some substances, such as wax, pitch and glass, which possess the outward 
appearance of being in the solid state, yield and flow under pressure, and they 
are sometimes regarded as highly viscous liquids. Solids may be crystalline or 
amorphous, and the crystalline state differs from the amorphous state in the 
regular arrangement of the constituent molecules, atoms or ions into some fixed 
and rigid pattern known as a lattice. Actually, many of the substances that were 
once considered to be amorphous have now been shown, by X-ray analysis, to 
exhibit some degree of regular molecular arrangement, but the term 'crystalline' 
is most frequently used to indicate a high degree of internal regularity, resulting 
in the development of definite external crystal faces. 

As molecular motion in a gas or liquid is free and random, the physical 
properties of these fluids are the same no matter in what direction they are 
measured. In other words, they are isotropic. True amorphous solids, because 
of the random arrangement of their constituent molecules, are also isotropic. 
Most crystals, however, are anisotropic; their mechanical, electrical, magnetic 
and optical properties can vary according to the direction in which they are 
measured. Crystals belonging to the cubic system are the exception to this rule; 
their highly symmetrical internal arrangement renders them optically isotropic. 
Anisotropy is most readily detected by refractive index measurements, and the 
striking phenomenon of double refraction exhibited by a clear crystal of Iceland 
spar (calcite) is probably the best-known example. 

1.1 Liquid crystals 

Before considering the type of crystal with which everyone is familiar, namely 
the solid crystalline body, it is worth while mentioning a state of matter which 
possesses the flow properties of a liquid yet exhibits some of the properties of 
the crystalline state. 

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2 Crystallization 

Although liquids are usually isotropic, some 200 cases are known of sub- 
stances that exhibit anisotropy in the liquid state at temperatures just above 
their melting point. These liquids bear the unfortunate, but popular, name 
'liquid crystals': the term is inapt because the word 'crystal' implies the exist- 
ence of a precise space lattice. Lattice formation is not possible in the liquid 
state, but some form of molecular orientation can occur with certain types of 
molecules under certain conditions. Accordingly, the name 'anisotropic liquid' 
is preferred to 'liquid crystal'. The name 'mesomorphic state' is used to indicate 
that anisotropic liquids are intermediate between the true liquid and crystalline 
solid states. 

Among the better-known examples of anisotropic liquids are />-azoxyphene- 
tole, />-azoxyanisole, cholesteryl benzoate, ammonium oleate and sodium 
stearate. These substances exhibit a sharp melting point, but they melt to form 
a turbid liquid. On further heating, the liquid suddenly becomes clear at some 
fixed temperature. On cooling, the reverse processes occur at the same tem- 
peratures as before. It is in the turbid liquid stage that anisotropy is exhibited. 
The changes in physical state occurring with change in temperature for the case 
of /?-azoxyphenetole are: 

.. . 137°C , . , ,. . , 167°C , t . . , 

solid , turbid liquid , clear liquid 

(anisotropic) (anisotropic, (isotropic) 


The simplest representation of the phenomenon is given by Bose's swarm 
theory, according to which molecules orientate into a number of groups in 
parallel formation (Figure 1.1). In many respects this is rather similar to the 
behaviour of a large number of logs floating down a river. Substances that can 
exist in the mesomorphic state are usually organic compounds, often aromatic, 
with elongated molecules. 

The mesomorphic state is conveniently divided into two main classes. The 
smectic (soap-like) state is characterized by an oily nature, and the flow of such 
liquids occurs by a gliding movement of thin layers over one another. Liquids in 
the nematic (thread-like) state flow like normal viscous liquids, but mobile 
threads can often be observed within the liquid layer. A third class, in which 


?$%$ M^mmi 

// v 

/i':>/vVi/ N 

(a) (b) 

Figure 1.1, Isotropic and anisotropic liquids, (a) Isotropic: molecules in random arrange- 
ment; (b) anisotropic: molecules aligned into swarms 

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The crystalline state 3 

strong optical activity is exhibited, is known as the cholesteric state; some 
workers regard this state as a special case of the nematic. The name arises from 
the fact that cholesteryl compounds form the majority of known examples. 

For further information on this subject, reference should be made to the 
relevant references listed in the Bibliography at the end of this chapter. 

1.2 Crystalline solids 

The true solid crystal comprises a rigid lattice of molecules, atoms or ions, the 
locations of which are characteristic of the substance. The regularity of the 
internal structure of this solid body results in the crystal having a characteristic 
shape; smooth surfaces or faces develop as a crystal grows, and the planes of 
these faces are parallel to atomic planes in the lattice. Very rarely, however, do 
any two crystals of a given substance look identical; in fact, any two given 
crystals often look quite different in both size and external shape. In a way this 
is not very surprising, as many crystals, especially the natural minerals, have 
grown under different conditions. Few natural crystals have grown 'free'; most 
have grown under some restraint resulting in stunted growth in one direction 
and exaggerated growth in another. 

This state of affairs prevented the general classification of crystals for cen- 
turies. The first advance in the science of crystallography came in 1669 when 
Steno observed a unique property of all quartz crystals. He found that the angle 
between any two given faces on a quartz crystal was constant, irrespective of 
the relative sizes of these faces. This fact was confirmed later by other workers, 
and in 1784 Haiiy proposed his Law of Constant Interfacial Angles: the angles 
between corresponding faces of all crystals of a given substance are constant. 
The crystals may vary in size, and the development of the various faces (the 
crystal habit) may differ considerably, but the interfacial angles do not vary; 
they are characteristic of the substance. It should be noted, however, that 
substances can often crystallize in more than one structural arrangement (poly- 
morphism - see section 1.8) in which case Haiiy's law applies only to the 
crystals of a given polymorph. 

Interfacial angles on centimetre-sized crystals, e.g. geological specimens, may 
be measured with a contact goniometer, consisting of an arm pivoted on a 
protractor {Figure 1.2), but precisions greater than 0.5° are rarely possible. The 
reflecting goniometer (Figure 1.3) is a more versatile and accurate apparatus. A 
crystal is mounted at the centre of a graduated turntable, a beam of light from 
an illuminated slit being reflected from one face of the crystal. The reflection is 
observed in a telescope and read on the graduated scale. The turntable is then 
rotated until the reflection from the next face of the crystal is observed in the 
telescope, and a second reading is taken from the scale. The difference a 
between the two readings is the angle between the normals to the two faces, 
and the interfacial angle is therefore (180 — a)° . 

Modern techniques of X-ray crystallography enable lattice dimensions and 
interfacial angles to be measured with high precision on milligram samples of 
crystal powder specimens. 

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Figure 1.2. Simple contact goniometer 


Slit and 


Figure 1.3. Reflecting goniometer 

1.3 Crystal symmetry 

Many of the geometric shapes that appear in the crystalline state are readily 
recognized as being to some degree symmetrical, and this fact can be used as 
a means of crystal classification. The three simple elements of symmetry which 
can be considered are: 

1. Symmetry about a point (a centre of symmetry) 

2. Symmetry about a line (an axis of symmetry) 

3. Symmetry about a plane (a plane of symmetry) 

It must be remembered, however, that while some crystals may possess a centre 
and several different axes and planes of symmetry, others may have no element 
of symmetry at all. 

A crystal possesses a centre of symmetry when every point on the surface of 
the crystal has an identical point on the opposite side of the centre, equidistant 
from it. A perfect cube is a good example of a body having a centre of 
symmetry (at its mass centre). 

If a crystal is rotated through 360° about any given axis, it obviously returns to 
its original position. If, however, the crystal appears to have reached its original 

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The crystalline state 

6 digd oxes 4 triad axes 3 tetrad axes 

Figure 1.4. The 13 axes of symmetry in a cube 


3 rectangular 

6 diagonal 

Figure 1.5. The 9 planes of symmetry in a cube 

position more than once during its complete rotation, the chosen axis is an axis 
of symmetry. If the crystal has to be rotated through 180° (360/2) before 
coming into coincidence with its original position, the axis is one of twofold 
symmetry (called a diad axis). If it has to be rotated through 120° (360/3), 90° 
(360/4) or 60° (360/6) the axes are of threefold symmetry (triad axis), fourfold 
symmetry (tetrad axis) and sixfold symmetry (hexad axis), respectively. These 
are the only axes of symmetry possible in the crystalline state. 

A cube, for instance, has 13 axes of symmetry: 6 diad axes through opposite 
edges, 4 triad axes through opposite corners and 3 tetrad axes through opposite 
faces. One each of these axes of symmetry is shown in Figure 1.4. 

The third simple type is symmetry about a plane. A plane of symmetry 
bisects a solid object in such a manner that one half becomes the mirror image 
of the other half in the given plane. This type of symmetry is quite common and 
is often the only type exhibited by a crystal. A cube has 9 planes of symmetry: 3 
rectangular planes each parallel to two faces, and 6 diagonal planes passing 
through opposite edges, as shown in Figure 1.5. 

It can be seen, therefore, that the cube is a highly symmetrical body, as it 
possesses 23 elements of symmetry (a centre, 9 planes and 13 axes). An octa- 
hedron also has the same 23 elements of symmetry; so, despite the difference 
in outward appearance, there is a definite crystallographic relationship between 
these two forms. Figure 1.6 indicates the passage from the cubic (hexahedral) to 
the octahedral form, and vice versa, by a progressive and symmetrical removal 
of the corners. The intermediate solid forms shown (truncated cube, truncated 
octahedron and cubo-octahedron) are three of the 13 Archimedean semi- 
regular solids which are called combination forms, i.e. combinations of a cube 
and an octahedron. Crystals exhibiting combination forms are commonly 
encountered (see Figure 1.20). 

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Truncated cube 

Truncated octahedron 

Cube Octahedron 

Figure 1.6. Combination forms of cube and octahedron 

The tetrahedron is also related to the cube and octahedron; in fact these three 
forms belong to the five regular solids of geometry. The other two (the regular 
dodecahedron and icosahedron) do not occur in the crystalline state. The 
rhombic dodecahedron, however, is frequently found, particularly in crystals 
of garnet. Table 1.1 lists the properties of the six regular and semi-regular forms 
most often encountered in crystals. The Euler relationship is useful for calcu- 
lating the number of faces, edges and corners of any polyhedron: 

E =F+C-2 

This relationship states that the number of edges is two less than the sum of the 
number of faces and corners. 

A fourth element of symmetry which is exhibited by some crystals is known 
by the names 'compound, or alternating, symmetry', or symmetry about a 

Table 1.1. Properties of some regular and semi-regular forms found in the crystalline state 





Edges at 


ts of symmetry 

a corner 




Regular solids 









Hexahedron (cube) 
















Semi-regular solids 

Truncated cube 








Truncated octahedron 
















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The crystalline state 

Figure 1.7. An axis of compound symmetry 

'rotation-reflection axis' or 'axis of rotatory inversion'. This type of symmetry 
obtains when one crystal face can be related to another by performing two 
operations: (a) rotation about an axis, and (b) reflection in a plane at right 
angles to the axis, or inversion about the centre. Figure 7.7 illustrates the case of 
a tetrahedron, where the four faces are marked A, B, C and D. Face A can be 
transformed into face B after rotation through 90°, followed by an inversion. 
This procedure can be repeated four times, so the chosen axis is a compound 
axis of fourfold symmetry. 

1.4 Crystal systems 

There are only 32 possible combinations of the above-mentioned elements of 
symmetry, including the asymmetric state (no elements of symmetry), and these 
are called the 32 point groups or classes. All but one or two of these classes have 
been observed in crystalline bodies. For convenience these 32 classes are 
grouped into seven systems, which are known by the following names: regular 
(5 possible classes), tetragonal (7), orthorhombic (3), monoclinic (3), triclinic 
(2), trigonal (5) and hexagonal (7). 

The first six of these systems can be described with reference to three axes, x, y 
and z. The z axis is vertical, and the x axis is directed from front to back and the 
y axis from right to left, as shown in Figure 1.8a. The angle between the axes y 
and z is denoted by a, that between x and z by [3, and that between x and y by 7. 
Four axes are required to describe the hexagonal system: the z axis is vertical 
and perpendicular to the other three axes (x, y and u), which are coplanar and 
inclined at 60° (or 120°) to one another, as shown in Figure 1.8b. Some workers 






Figure 1.8. Crystallographic axes for describing the seven crystal systems: (a) three axes 

yz = a; xz = /?; xy = 7; (b) four axes {hexagonal system) xy = yu = ux = 60° (120°) 

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8 Crystallization 

Table 1.2. The seven crystal systems 




Other names Angles between Length of Examples 

axes axes 




a — P — 7 = 90° x — y — z 

a = /? = 7 = 90° x = >• / z 

Orthorhombic Rhombic a — (5 — 7 = 90° i/;/z 




Monoclinic Monosymmetric a = /3 = 90° 7^ 7 i/;/z 

Triclinic Anorthic "/(3/7/ 90° i/)'/z 


Trigonal Rhombohedral a = (5 = 7 / 90° x — y — z 

Hexagonal None 

z axis is perpen- x = y — u ^ . 
dicular to the x, y 
and u axes, which 
are inclined at 60° 

Sodium chloride 


Nickel sulphate. 
7H 2 


Silver nitrate 

Potassium chlorate 
Oxalic acid 


Copper sulphate. 

5H 2 

Sodium nitrate 



Silver iodide 
Water (ice) 
Potassium nitrate 

prefer to describe the trigonal system with reference to four axes. Descriptions 
of the seven crystal systems, together with some of the other names occasionally 
employed, are given in Table 1.2. 

For the regular, tetragonal and orthorhombic systems, the three axes x, y and 
z are mutually perpendicular. The systems differ in the relative lengths of these 
axes: in the regular system they are all equal; in the orthorhombic system they 
are all unequal; and in the tetragonal system two are equal and the third is 
different. The three axes are all unequal in the monoclinic and triclinic systems; 
in the former, two of the angles are 90° and one angle is different, and in the 
latter all three angles are unequal and none is equal to 90°. Sometimes the 
limitation 'not equal to 30°, 60° or 90°' is also applied to the triclinic system. In 
the trigonal system three equal axes intersect at equal angles, but the angles are 

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The crystalline state 

Tetrahedron Sphenoid 

Figure 1.9. Hemihedral forms of the octahedron and tetragonal hipyramid 

not 90°. The hexagonal system is described with reference to four axes. The axis 
of sixfold symmetry (hexad axis) is usually chosen as the z axis, and the other 
three equal-length axes, located in a plane at 90° to the z axis, intersect one 
another at 60° (or 120°). 

Each crystal system contains several classes that exhibit only a partial sym- 
metry; for instance, only one-half or one-quarter of the maximum number of 
faces permitted by the symmetry may have been developed. The holohedral 
class is that which has the maximum number of similar faces, i.e. possesses the 
highest degree of symmetry. In the hemihedral class only half this number of 
faces have been developed, and in the tetrahedral class only one-quarter have 
been developed. For example, the regular tetrahedron (4 faces) is the hemi- 
hedral form of the holohedral octahedron (8 faces) and the wedge-shaped 
sphenoid is the hemihedral form of the tetragonal bipyramid {Figure 1.9). 

It has been mentioned above that crystals exhibiting combination forms are 
often encountered. The simplest forms of any crystal system are the prism and 
the pyramid. The cube, for instance, is the prism form of the regular system and 
the octahedron is the pyramidal form, and some combinations of these two 
forms have been indicated in Figure L6. Two simple combination forms in 
the tetragonal system are shown in Figure 1.10. Figures 1.10a and b are the 
tetragonal prism and bipyramid, respectively. Figure 1.10c shows a tetragonal 
prism that is terminated by two tetragonal pyramids, and Figure l.lOd the 







Figure 1.10. Simple combination forms in the tetragonal system: (a) tetragonal prism; 
(b) tetragonal bipyramid; (c) combination of prism and bipyramid; (d) combination of two 

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combination of two different tetragonal bipyramids. It frequently happens that 
a crystal develops a group of faces which intersect to form a series of parallel 
edges: such a set of faces is said to constitute a zone. In Figure 1.10b, for 
instance, the four prism faces make a zone. 

The crystal system favoured by a substance is to some extent dependent on 
the atomic or molecular complexity of the substance. More than 80 per cent of 
the crystalline elements and very simple inorganic compounds belong to the 
regular and hexagonal systems. As the constituent molecules become more 
complex, the orthorhombic and monoclinic systems are favoured; about 80 
per cent of the known crystalline organic substances and 60 per cent of the 
natural minerals belong to these systems. 

1.5 Miller indices 

All the faces of a crystal can be described and numbered in terms of their axial 
intercepts. The axes referred to here are the crystallographic axes (usually three) 
which are chosen to fit the symmetry; one or more of these axes may be axes of 
symmetry or parallel to them, but three convenient crystal edges can be used if 
desired. It is best if the three axes are mutually perpendicular, but this cannot 
always be arranged. On the other hand, crystals of the hexagonal system are 
often allotted four axes for indexing purposes. 

If, for example, three crystallographic axes have been decided upon, a plane 
that is inclined to all three axes is chosen as the standard or parametral plane. It 
is sometimes possible to choose one of the crystal faces to act as the parametral 
plane. The intercepts X, Y and Z of this plane on the axes x, y and z are called 
parameters a, b and c. The ratios of the parameters a : b and b : c are called the 
axial ratios, and by convention the values of the parameters are reduced so that 
the value of b is unity. 

W. H. Miller suggested, in 1839, that each face of a crystal could be repres- 
ented by the indices h, k and /, defined by 

h = — , k = — and I = — 
X Y Z 



_ N. 



6/ / 7 

1 •/ 1 / 

X* , 




+ y 

Figure 1.11. Intercepts of planes on the crystallographic axes 

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The crystalline state 


For the parametral plane, the axial intercepts X, Y and Z are the parameters a, 
b and c, so the indices h, k and / are a/ a, b/b and c/c, i.e. 1, 1 and 1. This is 
usually written (1 1 1). The indices for the other faces of the crystal are calculated 
from the values of their respective intercepts X, Y and Z, and these intercepts 
can always be represented by ma, nb and pc, where m, n and p are small whole 
numbers or infinity (Hauy's Law of Rational Intercepts). 

The procedure for allotting face indices is indicated in Figure 1.11, where 
equal divisions are made on the x, y and z axes. The parametral plane ABC, 
with axial intercepts of OA = a, OB = b and OC = c, respectively, is indexed 
(111), as described above. Plane DEF has axial intercepts X = OD = 2a, 
Y = OE = 3b and Z = OF = 3c; so the indices for this face can be calculated 

h = ajX = a/2a 

Ic = b/Y = bl3b = - 

l = cjZ = cj3c = - 

Hence h : k : / = \ : \ : I, and multiplying through by six, h:k:l= 3:2:2. Face 
DEF, therefore, is indexed (322). Similarly, face DFG, which has axial inter- 
cepts of X = 2a, Y = —2b and Z = 3c, gives h : k : I = j : — ^ : ^ = 3 : —3 : 2 or 
(332). Thus the Miller indices of a face are inversely proportional to its axial 

The generally accepted notation for Miller indices is that (hkl) represents a 
crystal face or lattice plane, while {hkl} represents a crystallographic form 
comprising all faces that can be derived from hkl by symmetry operations of 
the crystal. 

Figure 1.12 shows two simple crystals belonging to the regular system. As 
there is no inclined face in the cube, no face can be chosen as the parametral 
plane (111). The intercepts Y and Z of face A on the axes y and z are at infinity, 



Figure 1.12. Two simple crystals belonging to the regular system, showing the use of Miller 

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so the indices h, k and / for this face will be a/a, b/oo and c/oo, or (100). 
Similarly, faces B and C are designated (010) and (001), respectively. For the 
octahedron, face A is chosen arbitrarily as the parametral plane, so it is 
designated (111). As the crystal belongs to the regular system, the axial inter- 
cepts made by the other faces are all equal in magnitude, but not in sign, to the 
parametral intercepts a, b and c. For instance, the intercepts of face B on the z 
axis is negative, so this face is designated (111). Similarly, face C is designated 
(ill), and the unmarked D face is (ill). 

Figure 1.13 shows some geometrical figures representing the seven crystal 
systems, and Figure 1.14 indicates a few characteristic forms exhibited by 
crystals of some common substances. 

Occasionally, after careful goniometric measurement, crystals may be found 
to exhibit plane surfaces which appear to be crystallographic planes, being 
symmetrical in accordance with the symmetry of the crystal, but which cannot 
be described by simple indices. These are called vicinal faces. A simple method 
for determining the existence of these faces is to observe the reflection of a spot 
of light on the face: four spot reflections, for example, would indicate four 
vicinal faces. 

The number of vicinal faces corresponds to the symmetry of the face, and this 
property may often be used as an aid to the classification of the crystal. For 
example, a cube face (fourfold axis of symmetry) may appear to be made up of 
an extremely flat four-sided pyramid with its base being the true (100) plane but 
its apex need not necessarily be at the centre of the face. An octahedral face 
(threefold symmetry) may show a three-sided pyramid. These vicinal faces most 
probably arise from the mode of layer growth on the individual faces commen- 
cing at point sources (see section 6.1). 
















a=/3=90°* y a»/3*y*90° a=/2.=x.*90° 
Figure 1.13. The seven crystal systems 

see Table 

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The crystalline state 13 

III 001 




101 / 






* 001 





Sucrose (monoclinic) 

Copper sulphate ttriclinic) 

121- - 




Oio w 001 
Colcite (trigonal) 


Ammonium sulphate 






Sodium thiosulphote 

Figure 1.14. Some characteristic crystal forms 

101 001 


101 102 I,] 

Sodium chlorate 

1.6 Space lattices 

The external development of smooth faces on a crystal arises from some 
regularity in the internal arrangement of the constituent ions, atoms or mole- 
cules. Any account of the crystalline state, therefore, should include some 
reference to the internal structure of crystals. It is beyond the scope of this 
book to deal in any detail with this large topic, but a brief description will be 
given of the concept of the space lattice. For further information reference 
should be made to the specialized works listed in the Bibliography. 

It is well known that some crystals can be split by cleavage into smaller 
crystals which bear a distinct resemblance in shape to the parent body. While 
there is clearly a mechanical limit to the number of times that this process can 
be repeated, eighteenth century investigators, Hooke and Haiiy in particular, 
were led to the conclusion that all crystals are built up from a large number of 
minute units, each shaped like the larger crystal. This hypothesis constituted a 
very important step forward in the science of crystallography because its logical 
extension led to the modern concept of the space lattice. 

A space lattice is a regular arrangement of points in three dimensions, each 
point representing a structural unit, e.g. an ion, atom or a molecule. The whole 

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Table 1.3. The fourteen Bravais lattices 

Type of symmetry 








crystal system 


Body-centred cube 
Face-centred cube 

Square prism 
Body-centred square prism 

Rectangular prism 
Body-centred rectangular prism 
Rhombic prism 
Body-centred rhombic prism 

Monoclinic parallelepiped 
Clinorhombic prism 

Triclinic parallelepiped 


Hexagonal prism 





structure is homogeneous, i.e. every point in the lattice has an environment 
identical with every other point's. For instance, if a line is drawn between any 
two points, it will, when produced in both directions, pass through other points 
in the lattice whose spacing is identical with that of the chosen pair. Another 
way in which this homogeneity can be visualized is to imagine an observer 
located within the structure; he would get the same view of his surroundings 
from any of the points in the lattice. 

By geometrical reasoning, Bravais postulated in 1848 that there were only 14 
possible basic types of lattice that could give the above environmental identity. 
These 14 lattices can be classified into seven groups based on their symmetry, 
which correspond to the seven crystal systems listed in Table 1.2. The 14 
Bravais lattices are given in Table 1.3. The three cubic lattices are illustrated 
in Figure 1.15; the first comprises eight structural units arranged at the corners 
of a cube, the second consists of a cubic structure with a ninth unit located at 
the centre of the cube, and the third of a cube with six extra units each located 
on a face of the cube. 

(a) (b) (c) 

Figure 1.15. The three cubic lattices: (a) cube; (b) body-centred cube; (c) face-centred cube 

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The crystalline state 1 5 

The points in any lattice can be arranged to lie on a larger number of 
different planes, called lattice planes, some of which will contain more points 
per unit area than others. The external faces of a crystal are parallel to lattice 
planes, and the most commonly occurring faces will be those which correspond 
to planes containing a high density of points, usually referred to as a high 
reticular density (Law of Bravais). Cleavage also occurs along lattice planes. 
Bravais suggested that the surface energies, and hence the rates of growth, 
should be inversely proportional to the reticular densities, so that the planes of 
highest density will grow at the slowest rate and the low-density planes, by their 
high growth rate, may soon disappear. For these reasons, the shape of a grown 
crystal may not always reflect the symmetry expected from its basic unit cell 
(see section 6.4). 

Although there are only 14 basic lattices, interpenetration of lattices can 
occur in actual crystals, and it has been deduced that 230 combinations are 
posible which still result in the identity of environment of any given point. 
These combinations are the 230 space groups, which are divided into the 32 
point groups, or classes, mentioned above in connection with the seven crystal 
systems. The law of Bravais has been extended by Donnay and Harker in 1937 
into a more generalized form (the Bravais-Donnay-Harker Principle) by con- 
sideration of the space groups rather than the lattice types. 

1.7 Solid state bonding 

Four main types of crystalline solid may be specified according to the method 
of bonding in the solid state, viz. ionic, covalent, molecular and metallic. There 
are materials intermediate between these classes, but most crystalline solids can 
be classified as predominantly one of the basic types. 

The ionic crystals (e.g. sodium chloride) are composed of charged ions held in 
place in the lattice by electrostatic forces, and separated from the oppositely 
charged ions by regions of negligible electron density. In covalent crystals (e.g. 
diamond) the constituent atoms do not carry effective charges; they are con- 
nected by a framework of covalent bonds, the atoms sharing their outer 
electrons. Molecular crystals (e.g. organic compounds) are composed of dis- 
crete molecules held together by weak attractive forces (e.g. 7r-bonds or hydro- 
gen bonds). 

Metallic crystals (e.g. copper) comprise ordered arrays of identical cations. 
The constituent atoms share their outer electrons, but these are so loosely held 
that they are free to move through the crystal lattice and confer 'metallic' 
properties on the solid. For example, ionic, covalent and molecular crystals 
are essentially non-conductors of electricity, because the electrons are all locked 
into fixed quantum states. Metals are good conductors because of the presence 
of mobile electrons. 

Semiconducting crystals (e.g. germanium) are usually covalent solids with 
some ionic characteristics, although a few molecular solids (e.g. some polycyclic 
aromatic hydrocarbons such as anthracene) are known in which under certain 
conditions a small fraction of the valency electrons are free to move in the 

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16 Crystallization 

crystal. The electrical conductivity of semiconductors is electronic in nature, 
but it differs from that in metals. Metallic conductivity decreases when the 
temperature is raised, because thermal agitation exerts an impeding effect. On 
the other hand, the conductivity of a semiconductor increases with heating, 
because the number of electron-'hole' pairs, the electricity carriers in semicon- 
ductors, increases greatly with temperature. Metals have electrical resistivities 
in the ranges 10 -8 to 10" 6 S _1 m. Insulators cover the range 10 8 to 10 20 (dia- 
mond) and semiconductors 10 to 10 7 S _1 m. 

The electrical conductivity of a semiconductor can be profoundly affected by 
the presence of impurities. For example, if x silicon atoms in the lattice of a 
silicon crystal are replaced by x phosphorus atoms, the lattice will gain x 
electrons and a negative (n-type) semiconductor results. On the other hand, if 
x silicon atoms are replaced by x boron atoms, the lattice will lose x electrons 
and a positive (p-type) semiconductor is formed. The impurity atoms are called 
'donors' or 'acceptors' according to whether they give or take electrons to or 
from the lattice. 

1.8 Isomorphs and polymorphs 

Two or more substances that crystallize in almost identical forms are said to be 
isomorphous (Greek: 'of equal form'). This is not a contradiction of Haiiy's law, 
because these crystals do show small, but quite definite, differences in their 
respective interfacial angles. Isomorphs are often chemically similar and can 
then be represented by similar chemical formulae; this statement is one form of 
Mitscherlich's Law of Isomorphism, which is now recognized only as a broad 
generalization. One group of compounds which obey and illustrate Mitscher- 
lich's law is represented by the formula M 2 S0 4 • M" (S0 4 ) 3 • 24H 2 (the alums), 
where M' represents a univalent radical (e.g. K or NH4) and M'" represents a 
tervalent radical (e.g. Al, Cr or Fe). Many phosphates and arsenates, sulphates 
and selenates are also isomorphous. 

Sometimes isomorphous substances can crystallize together out of a solution 
to form 'mixed crystals' or, as they are better termed, crystalline 'solid solu- 
tions'. In such cases the composition of the homogeneous solid phase that is 
deposited follows no fixed pattern; it depends largely on the relative concentra- 
tions and solubilities of the substances in the original solvent. For instance, 
chrome alum, K2SO4 • Cr2(S04) 3 • 24H2O (purple), and potash alum, 
K2SO4 • Al2(S04) 3 • 24H2O (colourless), crystallize from their respective aque- 
ous solutions as regular octahedra. When an aqueous solution containing both 
salts is crystallized, regular octahedra are again formed, but the colour of the 
crystals (which are now homogeneous solid solutions) can vary from almost 
colourless to deep purple, depending on the proportions of the two alums in the 
crystallizing solution. 

Another phenomenon often shown by isomorphs is the formation of over- 
growth crystals. For example, if a crystal of chrome alum (octahedral) is placed 
in a saturated solution of potash alum, it will grow in a regular manner such 
that the purple core is covered with a continuous colourless overgrowth. In 

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The crystalline state 17 

a similar manner an overgrowth crystal of nickel sulphate, NiSC^ • 7H2O 
(green), and zinc sulphate, ZnSC>4 • 7H2O (colourless), can be prepared. 

There have been many 'rules' and 'tests' proposed for the phenomenon of 
isomorphism, but in view of the large number of known exceptions to these it is 
now recognized that the only general property of isomorphism is that crystals 
of the different substances shall show very close similarity. All the other proper- 
ties, including those mentioned above, are merely confirmatory and not neces- 
sarily shown by all isomorphs. 

A substance capable of crystallizing into different, but chemically identical, 
crystalline forms is said to exhibit polymorphism. Different polymorphs of a 
given substance are chemically identical but will exhibit different physical 
properties. Dimorphous and trimorphous substances are commonly known, 

Calcium carbonate: calcite (trigonal-rhombohedral) 
aragonite (orthorhombic) 
vaterite (hexagonal) 
Carbon: graphite (hexagonal) 
diamond (regular) 
Silicon dioxide: cristobalite (regular) 
tridymite (hexagonal) 
quartz (trigonal) 

The term allotropy instead of polymorphism is often used when the substance is 
an element. 

The different crystalline forms exhibited by one substance may result from a 
variation in the crystallization temperature or a change of solvent. Sulphur, for 
instance, crystallizes in the form of orthorhombic crystals (a-S) from a carbon 
disulphide solution, and of monoclinic crystals (J3-S) from the melt. In this 
particular case the two crystalline forms are interconvertible: /3-sulphur cooled 
below 95.5 °C changes to the a form. This interconversion between two crystal 
forms at a definite transition temperature is called enantiotropy (Greek: 'change 
into opposite') and is accompanied by a change in volume. 

Ammonium nitrate (melting point 169.6°C) exhibits five polymorphs and 
four enantiotropic changes between —18 and 125 °C, as shown below: 

(i) (ii) (in) 

liquid ■ cubic - trigonal ■ orthorhombic 

M 169.6°C 125.2°C 6 84.2°C 

(IV) (V) 

■ orthorhombic ■ tetragonal 

32.3°C -18°C b 

The transitions from forms II to III and IV to V result in volume increases: the 
changes from I to II and III to IV are accompanied by a decrease in volume. 
These volume changes frequently cause difficulty in the processing and storage 
of ammonium nitrate. The salt can readily burst a metal container into which it 
has been cast when change II to III occurs. The drying of ammonium nitrate 
crystals must be carried out within fixed temperature limits, e.g. 40-80 °C, 

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18 Crystallization 

otherwise the crystals can disintegrate when a transition temperature is 

Crystals of polymorphic substances sometimes undergo transformation with- 
out a change of external form, the result being an aggregate of very small 
crystals of the stable modification confined within the boundaries of the original 
unstable form. For example, an unstable rhombohedral form of potassium 
nitrate can crystallize from a warm aqueous solution, but when these crystals 
come into contact with a crystal of the stable modification, transformation 
sweeps rapidly through the rhombohedra which retain their shape. The crystals 
lose much of their transparency and acquire a finely granular appearance and 
their original mechanical strength is greatly reduced. Such pseudomorphs as they 
are called exhibit confused optical properties which cannot be correlated with 
the external symmetry (Hartshorne and Stuart, 1969). 

When polymorphs are not interconvertible, the crystal forms are said to be 
monotropic: graphite and diamond are monotropic forms of carbon. The term 
isopolymorphism is used when each of the polymorphous forms of one sub- 
stance is isomorphous with the respective polymorphous form of another 
substance. For instance, the regular and orthorhombic polymorphs of 
arsenious oxide, AS2O3, are respectively isomorphous with the regular and 
orthorhombic polymorphs of antimony trioxide, Sb203. These two oxides are 
thus said to be isodimorphous. 

Polytypism is a form of polymorphism in which the crystal lattice arrange- 
ments differ only in the manner in which identical two-dimensional arrays are 
stacked (Verma and Krishna, 1966). 

1.9 Enantiomorphs and chirality 

Isomeric substances, different compounds having the same formula, may be 
divided into two main groups: 

(a) constitutional isomers, which differ because their constituent atoms are 
connected in a different order, e.g., ethanol CH3CH2OH and dimethylether 
CH 3 OCH 3 , 

(b) stereoisomers, which differ only in the spacial arrangement of their con- 
stituent atoms. Stereoisomers can also be divided into two groups: 
(i) enantiomers, molecules that are mirror images of one another, and 
(ii) diastereomers, which are not. 

Diastereomers can have quite different properties, e.g., the cis- and trans- 
compounds maleic and fumaric acids which have different melting points, 
130 °C and 270 °C respectively. On the other hand, enantiomers have identical 
properties with one exception, viz., that of optical activity, the ability to rotate 
the plane of polarization of plane-polarized light. One form will rotate to the 
right {dextrorotatory) and the other to the left (laevorotatory). The direction 
and magnitude of rotation are measured with a polarimeter. 

Molecules and substances that exhibit optical activity are generally described 
as chiral (Greek cheir 'hand'). Two crystals of the same substance that are 

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The crystalline state 


mirror images of each other are said to be enantiomorphous (Greek 'of opposite 
form'). These crystals have neither planes of symmetry nor a centre of sym- 
metry. Enantiomorphous crystals are not necessarily optically active, but all 
known optically active substances are capable of being crystallized into enan- 
tiomorphous forms. In many cases the solution or melt of an optically active 
crystal is also optically active. However, if dissolution or melting destroys the 
optical activity, this is an indication that the molecular structure was not 

Tartaric acid (Figure 1.16) and certain sugars are well-known examples of 
optically active substances. Optical activity is generally associated with com- 
pounds that possess one or more atoms around which different elements or 
groups are arranged asymmetrically, i.e., a stereocentre, so that the molecule can 
exist in mirror image forms. The most common stereocentre in organic 
compounds is an asymmetric carbon atom, and tartaric acid offers a good 
example. Three possible arrangements of the tartaric acid molecule are shown 
in Figure 1.17. The (a) and (b) forms are mirror images of each other; both 
contain asymmetric carbon atoms and both are optically active; one will be the 
dextro-form and the other the laevo-form. Although there are two asymmetric 
carbon atoms in formula (c), this particular form (meso-tartaric acid) is optic- 
ally inactive; the potential optical activity of one-half of the molecule is com- 
pensated by the opposite potential optical activity of the other. 

Dextro- and laevo-forms are now designated in all modern texts as (+) and 
(— ) respectively. The optically inactive racemate, a true double compound 

ioi on ioi 

(a) (b) 

Figure 1.16. (a) Dextro- and (b) laevo-tartaric acid crystals (monoclinic system) 

OH-C— H H— C— OH 
H— C— OH OH— C— H 







H— C— OH 
H— C— OH 




Figure 1.17. The tartaric acid molecule: (a) and (b) optically active forms; (c) meso- 
tartaric acid, optically inactive 

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20 Crystallization 

(section 4.3.2), comprising an equimolar mixture of (+) and (— ) forms, is 
designated (±). The symbols d- and 1-, commonly found in older literature to 
designate optically active dextro- and laevo-forms, were abandoned to avoid 
confusion with the capital letters d and l which are still commonly used to 
designate molecular configuration, but not the direction of rotation of plane- 
polarized light. It is important to note, therefore, that not all d series com- 
pounds are necessarily dextrorotatory (+) nor are all l series compounds 
laevorotatory (— ). 

The d, l system, was arbitrarily based on the configuration of the enantio- 
meric glyceraldehyde molecules: the (+)-isomer was taken to have the structure 
implied by formula 1 and this arrangement of atoms was called the d configura- 
tion. Conversely, formula 2 was designated as representing the l configuration: 


H— C— OH HO— C— H 

CH 2 OH CH 2 OH 

(1) (2) 

Lactic acid provides a simple example of how the d, l system could be applied 
to other compounds. The relative configuration of lactic acid is determined by 
the fact that it can be synthesized from (D)-(+)-glyceraldehyde without break- 
ing any bonds to the asymmetric carbon atom: 


H— C— OH P- H— C— OH 

CH 2 OH CH 3 

(D) - (+) -glyceraldehyde (D) - (-)-lacticacid 

The lactic acid produced by this reaction, however, is laevorotatory not dextro- 
rotatory like the starting material, thus illustrating the above warning that there 
is no essential link between optical rotation and molecular configuration. 

The d, l system becomes ambiguous for all but the simplest of molecules and 
is now increasingly being replaced with the more logically based and adaptable 
r, s system, which has been internationally adopted by IUPAC for classifying 
absolute molecular configuration. Comprehensive accounts of the R, S conven- 
tion and its application are given in most modern textbooks on organic 
chemistry, but the following short introduction may serve as a brief guide to 
the procedure for classifying a compound with a single asymmetric carbon 
atom as the stereocentre. 

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The crystalline state 21 

First, the four different groups attached to the stereocentre are identified and 
each is assigned a priority number, 1 to 4, using the Cahn-Ingold-Prelog 
'sequence rules' according to which the highest priority (1) is given to the group 
with the atom directly attached to the stereocentre that has the highest atomic 
number. If by this rule two or more groups would at first appear to have 
identical priorities, the atomic numbers of the second atoms in each group 
are compared, continuing with the subsequent atoms until a difference is 
identified. For example, for a-aminopropionic acid {alanine) 


H— C — NH 2 


the following priorities would be assigned: NH2 = 1, COOH = 2, CH3 = 3 and 
H = 4. The model of the molecule is then oriented in space so that the stereo- 
centre is observed from the side opposite the lowest priority group. So observ- 
ing the stereocentre with the lowest priority group (H = 4) to the rear, the 
view would be 




NH 2 

I 4 V 


(2) (3) 


(3) (2) 

According to the Cahn-Ingold-Prelog rules, if the path from 1 to 2 to 3 runs 
clockwise the stereocentre is designated by the letter r (Latin: rectus, right). If 
the path runs anticlockwise it is designated by the letter s (Latin: sinister, left). 
If the structure has only one stereocentre, (r) or (s) is used as the first prefix to 
the name, e.g., (s)-aminopropionic acid. The optical rotation of the compound 
is indicated by a second prefix, e.g., (s)-(+)-aminopropionic acid, noting again 
as mentioned above for the d, l system, there is no necessary connection 
between (s) left and (r) right configurations and the (— ) left and (+) right 
directions of optical rotation. If the molecule has more than one stereocentre 
their designations and positions are identified in the prefix, e.g., (2r, 3r)- 

1.9.1 Racemism 

The case of tartaric acid serves to illustrate the property known as racemism. 
An equimolar mixture of crystalline d and l tartaric acids dissolved in water 
will produce an optically inactive solution. Crystallization of this solution will 

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22 Crystallization 

yield crystals of optically inactive racemic acid which are different in form from 
the d and l crystals. There is, however, a difference between a racemate and a 
mew-form of a substance; the former can be resolved into d and l forms but 
the latter cannot. 

Crystalline racemates are normally considered to belong to one of two basic 

1. Conglomerate: an equimolal mechanical mixture of two pure enantio- 

2. Racemic compound: an equimolal mixture of two enantiomers homo- 
geneously distributed throughout the crystal lattice. 

A racemate can be resolved in a number of ways. In 1848 Pasteur found that 
crystals of the sodium ammonium tartrate (racemate) 

Na • NH 4 • C 4 H 4 6 • H z O 

deposited from aqueous solution, consisted of two clearly different types, one 
being the mirror image of the other. The d and l forms were easily separated 
by hand picking. Although widely quoted, however, this example of manual 
resolution through visual observation is in fact a very rare occurrence. 

Bacterial attack was also shown by Pasteur to be effective in the resolution of 
racemic acid. Penicillium glaucum allowed to grow in a dilute solution of 
sodium ammonium racemate destroys the d form but, apart from being a 
rather wasteful process, the attack is not always completely selective. 

A racemate may also be resolved by forming a salt or ester with an optically 
active base (usually an amine) or alcohol. For example, a racemate of an acidic 
substance A with, say, the dextro form of an optically active base B will give 

dl.4 + dB — > dA ■ dB + lA ■ dB 

and the two salts dA ■ uB and lA ■ dB can then be separated by fractional 

A comprehensive account of the resolution of racemates is given by Jacques, 
Collet and Wilen (1981). This topic is further discussed in section 7.2. 

1.10 Crystal habit 

Although crystals can be classified according to the seven general systems 
(Table 1.1), the relative sizes of the faces of a particular crystal can vary 
considerably. This variation is called a modification of habit. The crystals 
may grow more rapidly, or be stunted, in one direction; thus an elongated 
growth of the prismatic habit gives a needle-shaped crystal (acicular habit) and 
a stunted growth gives a flat plate-like crystal (tabular, platy or flaky habit). 
Nearly all manufactured and natural crystals are distorted to some degree, and 
this fact frequently leads to a misunderstanding of the term 'symmetry'. Perfect 
geometric symmetry is rarely observed in crystals, but crystallographic sym- 
metry is readily detected by means of a goniometer. 

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The crystalline state 







(a) Tabular (b) Prismatic (c) Acicular 

Figure 1.18. Crystal habit illustrated on a hexagonal crystal 

Figure 1.19. Some common habits of potassium sulphate crystals (orthorhombic system): 
a = {100},/) = {010}, c= {011},/= {021},m = {110}, o = {111}, / = {130} 

Figure 1.18 shows three different habits of a crystal belonging to the hexa- 
gonal system. The centre diagram (b) shows a crystal with a predominant 
prismatic habit. This combination-form crystal is terminated by hexagonal 
pyramids and two flat faces perpendicular to the vertical axis; these flat parallel 
faces cutting one axis are called pinacoids. A stunted growth in the vertical 
direction (or elongated growth in the directions of the other axes) results in a 
tabular crystal (a); excessively flattened crystals are usually called plates or 
flakes. An elongated growth in the vertical direction yields a needle or acicular 
crystal (c); flattened needle crystals are often called blades. 

Figure 1.19 shows some of the habits exhibited by potassium sulphate crys- 
tals grown from aqueous solution and Figure 1.20 shows four different habits of 
sodium chloride crystals. 

The relative growths of the faces of a crystal can be altered, and often 
controlled, by a number of factors. Rapid crystallization, such as that produced 
by the sudden cooling or seeding of a supersaturated solution, may result in the 
formation of needle crystals; impurities in the crystallizing solution can stunt 

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24 Crystallization 

Figure 1.20. Four different habits of sodium chloride {regular system) crystals (Courtesy of 
ICI Ltd, Mond Division) 

the growth of a crystal in certain directions; and crystallization from solutions 
of the given substance in different solvents generally results in a change of 
habit. The degree of supersaturation or supercooling of a solution or melt often 
exerts a considerable influence on the crystal habit, and so can the state of 
agitation of the system. These and other factors affecting the control of crystal 
habit are discussed in section 6.4. 

1.11 Dendrites 

Rapid crystallization from supercooled melts, supersaturated solutions and 
vapours frequently produces tree-like formations called dendrites, the growth 
of which is indicated in Figure 1.21. The main crystal stem grows quite rapidly 
in a supercooled system that has been seeded, and at a later stage primary 
branches grow at a slower rate out of the stem, often at right angles to it. In 
certain cases, small secondary branches may grow slowly out of the primaries. 
Eventually branching ceases and the pattern becomes filled in with crystalline 

Most metals crystallize from the molten state in this manner, but because of 
the filling-in process the final crystalline mass may show little outward appear- 

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The crystalline state 





_Lu Ll^L. 

mi 'I ' : 
, i I .. 1 1 

H. . I , I ,.l ; I 

1 ' ' iyi 'im ' I 
Primary branch 

pn-T — r 

2j LlJ_ 

"I ' 

■'■ I » ' I ■ 



Main Secondary branch 

Figure 1.21. Dendritic growth 

ance of dendrite formation. The fascinating patterns of snow crystals are good 
examples of dendritic growth, and the frosting of windows often affords a 
visual observation of this phenomenon occurring in two dimensions. The 
growth of a dendrite can be observed quite easily under a microscope by 
seeding a drop of a supersaturated solution on the slide. 

Dendrites form most commonly during the early stages of crystallization; at 
later stages a more normal uniform growth takes place and the pattern may be 
obliterated. Dendritic growth occurs quite readily in thin liquid layers, prob- 
ably because of the high rate of evaporative cooling, whereas agitation tends to 
suppress this type of growth. Dendrite formation tends to be favoured by 
substances that have a high enthalpy of crystallization and a low thermal 

1.12 Composite crystals and twins 

Most crystalline natural minerals, and many crystals produced industrially, 
exhibit some form of aggregation or intergrowth, and prevention of the forma- 
tion of these composite crystals is one of the problems of large-scale crystal- 
lization. The presence of aggregates in a crystalline mass spoils the appearance 
of the product and interferes with its free-flowing nature. More important, 
however, aggregation is often indicative of impurity because crystal clusters 
readily retain impure mother liquor and resist efficient washing (section 9.7.2). 
Composite crystals may occur in simple symmetrical forms or in random 
clusters. The simplest form of aggregate results from the phenomenon known 
as parallel growth; individual forms of the same substance grow on the top of 
one another in such a manner that all corresponding faces and edges of the 
individuals are parallel. Potash alum, K2SO4 • A^SO^ • 24H2O, exhibits this 
type of growth; Figure 1.22 shows a typical structure in which regular octahedra 

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26 Crystallization 

Figure 1.22. Parallel growth on a crystal of potash alum 

Figure 1.23. Interpenetrant twin of two cubes (e.g. fluorspar) 

are piled on top of one another in a column symmetrical about the vertical 
axis. Parallel growth is often associated with isomorphs; for instance, parallel 
growths of one alum can be formed on the crystals of another, but this property 
is no longer regarded as an infallible test for isomorphism. 

Another composite crystal frequently encountered is known as a twin or 
a made; it appears to be composed of two intergrown individuals, similar in 
form, joined symmetrically about an axis (a twin axis) or a plane (a twin plane). 
A twin axis is a possible crystal edge and a twin plane is a possible crystal face. 
Many types of twins may be formed in simple shapes such as a V, +, L and so 
forth, or they may show an interpenetration giving the appearance of one 
individual having passed completely through the other (Figure 1.23). Partial 
interpenetration (Figure 1.24) can also occur. In some cases, a twin crystal may 
present the outward appearance of a form that possesses a higher degree of 
symmetry than that of the individuals, and this is known as mimetic twinning. 
A typical example of this behaviour is orthorhombic potassium sulphate, which 
can form a twin looking almost identical with a hexagonal bipyramid. 

Parallel growth and twinning (or even triplet formation) are usually encoun- 
tered when crystallization has been allowed to take place in an undisturbed 
medium. Although twins of individuals belonging to most of the seven crystal 
systems are known, twinning occurs most frequently when the crystals belong 
to the orthorhombic or monoclinic systems. Certain impurities in the crystal- 
lizing medium can cause twin formation even under vigorously agitated condi- 
tions: this is one of the problems encountered in the commercial crystallization 
of sugar. 

The formation of crystal clusters, aggregates or conglomerates which possess 
no symmetrical properties is probably more frequently encountered in large- 
scale crystallization than the formation of twins. Relatively little is still known 

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The crystalline state 


Figure 1.24. Partial interpenetrant twin {e.g. quartz) 

about the growth of these irregular crystal masses, but among the factors that 
generally favour their formation are poor agitation, the presence of certain 
impurities in the crystallizing solution, seeding at high degrees of supersatura- 
tion and the presence of too many seed crystals, leading to conditions of 
overcrowding in the crystallizer. 

1.13 Imperfections in crystals 

Very few crystals are perfect. Indeed, in many cases they are not required to be, 
since lattice imperfections and other defects can confer some important chem- 
ical and mechanical properties on crystalline materials. Surface defects can also 
greatly influence the process of crystal growth. There are three main types of 
lattice imperfection: point (zero-dimensional, line (one-dimensional) and sur- 
face (two-dimensional). 

1.13.1 Point defects 

The common point defects are indicated in Figure 1.25. Vacancies are lattice 
sites from which units are missing, leaving 'holes' in the structure. These units 
may be atoms, e.g. in metallic crystals, molecules (molecular crystals) or ions 
(ionic crystals). The inter stitials are foreign atoms that occupy positions in the 
interstices between the matrix atoms of the crystal. In most cases the occurrence 
of interstitials leads to a distortion of the lattice. 

More complex point defects can occur in ionic crystals. For example, a cation 
can leave its site and become relocated interstitially near a neighbouring cation. 
This combination of defects (a cation vacancy and an interstitial cation) is 
called a Frenkel imperfection. A cation vacancy combined with an anion 
vacancy is called a Schottky imperfection. 

A foreign atom that occupies the site of a matrix atom is called a substitutional 
impurity. Many types of semiconductor crystals contain controlled quantities of 
substitutional impurities. Germanium crystals, for example, can be grown con- 
taining minute quantities of aluminium (p-type semiconductors) or phosphorus 

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28 Crystallization 

Figure 1.25. Representation of some common point defects: A, interstitial impurity; 
B, substitutional impurity; C, vacancy 

1.13.2 Line defects 

The two main types of line defect which can play an important role in the model 
of crystal growth are the edge and screw dislocations. Both of these are respon- 
sible for slip or shearing in crystals. Large numbers of dislocations occur in 
most crystals; they form readily during the growth process under the influence 
of surface and internal stresses. 

Figure 1.26 shows in diagrammatic form the cross-sectional view of a crystal 
lattice in which the lower part of a vertical row of atoms is missing. The 
position of the dislocation is marked by the symbol _l_; the vertical stroke of 
this symbol indicates the extra plane of atoms and the horizontal stroke 
indicates the slip plane. The line passing through all the points _L, i.e. drawn 
vertical to the plane of the diagram, is called the edge dislocation line. In an 
edge dislocation, therefore, the atoms are displaced at right angles to the 
dislocation line. 

The process of slip under the action of a shearing force may be explained as 
follows (see Figure 1.26). The application of a shear stress to a crystal causes 
atom A to move further away from atom B and closer to atom C. The bond 
between A and B, which is already strained, breaks and a new bond is formed 
between A and C. The dislocation thus moves one atomic distance to the right, 
and if this process is continued the dislocation will eventually reach the edge of 
the crystal. The direction and magnitude of slip are indicated by the Burgers 
vector, which may be one or more atomic spacings. In the above example, 
where the displacement is one lattice spacing, the Burgers vector is equal to 1 . 

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The crystalline state 29 

Direction of 

— T~ "/ T / / — Slip plane 

_Direction of 

Figure 1.26. Movement of an edge dislocation through a crystal 

Figure 1.27. A screw dislocation 

A screw dislocation forms when the atoms are displaced along the dis- 
location line, rather than at right angles to it as in the case of the edge 
dislocation. Figure 1.27 indicates this type of lattice distortion. In this example 
the Burgers vector is 1 (unit step height), but its magnitude may be any integral 

Screw dislocations give rise to a particular mode of growth in which the 
attachment of growth units to the face of the dislocation results in the devel- 
opment of a spiral growth pattern over the crystal face (see section 6.1.2). 


-[1-31/31] 9.3.2001 12:05PM 




Figure 1.28. A simple tilt boundary 

1.13.3 Surface defects 

A variety of surface imperfections, or mismatch boundaries, can be produced in 
crystalline materials as a result of mechanical or thermal stresses or irregular 
growth. Grain boundaries, for example, can be created between individual 
crystals of different orientation in a polycrystalline aggregate. 

When the degree of mismatching is small, the boundary can be considered to 
be composed of a line of dislocations. A low-angle tilt boundary is equivalent to 
a line of edge dislocations, and the angle of tilt is given by 9 = b/h where b is the 
Burgers vector and h the average vertical distance between the dislocations 
(Figure 1.28). A twist boundary can be considered, when the degree of twist is 
small, as a succession of parallel screw dislocations. For a full account of this 
subject reference should be made to the specialized works (see Bibliography). 


Buckley, H. E. (1952) Crystal Growth. London: Wiley 

Bunn, C. W. (1961) Chemical Crystallography, 2nd edn. Oxford: Clarendon Press 
Chandrasekhar, S. (1992) Liquid Crystals, 2nd edn. Cambridge: University Press 
Hartshorne, N. H. and Stuart, A. (1969) Practical Optical Crystallography, 2nd edn. 

London: Arnold 
Hirth, J. P. and Loethe, J. (1968) Theory of Dislocations. New York: McGraw-Hill 
Kelly, A. and Groves, G. (2000) Crystallography and Crystal Defects. Chichester: Wiley 

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The crystalline state 3 1 

McKie, D. and McKie, C. (1974) Crystalline Solids, 3rd edn. London: Nelson 
Phillips, F. C. (1980) Introduction to Crystallography, 4th edn. London: Longmans 

Rosenberg, H. M. (1978) The Solid State. Oxford: Clarendon Press 
Vainshtein, B. K., Chernov, A. A. and Shuvalov, L. A. (1984) (eds.) Modern Crystal- 
lography, vol. 1 Symmetry of Crystals; vol. 2 Structure of Crystals; vol. 3 Crystal 
Growth; vol. 4 Physical Properties of Crystals. Berlin: Springer 
Van Bueren, H. G. (1960) Imperfections in Crystals. New York: Interscience 
Wells, A. F. (1962) Structural Inorganic Chemistry, 3rd edn. Oxford: Clarendon Press 
Wright, J. D. (1987) Molecular Crystals. Cambridge: University Press 

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2 Physical and thermal properties 

2.1 Density 

2.1.1 Solids 

The densities of most pure solid substances are readily available in the standard 
physical property handbooks. The densities of actual crystallized substances, 
however, may differ from the literature values on account of the presence of 
vapour or liquid inclusions (see section 6.6) or adhering surface moisture. For 
example, recorded values of 'commercially pure' sucrose crystals have ranged 
from 1580 to 1610 kg m -3 compared with an expected value of 1587kgm~ 3 . 

The theoretical density, p*, of a crystal may be calculated from the lattice 
parameters (section 1.6) by means of the relationship: 

* nM n n 

P ° = VN (2A) 

Where n is the number of formula units in the unit cell, V is the volume of the 
unit cell, M is the molar mass of the substance and N is the Avogadro number 
(6.023 x 10 26 kmol _1 ). For sucrose, the lengths of the a, b and c axes of this 
monoclinic crystal are 10.9, 8.70 and 7.75 x 10~ 10 m, respectively, with the 
angle /3 = 103° (sin (3 = 0.9744). Hence the volume V = 716.1 x 10- 30 m 3 . 
Further, M = 342.3 kg kmol~' and n = 2. Substituting these values in equation 
2.1 gives a value of p* = 1587kgm~ 3 . 

The actual density of a solid substance, even of a relatively small individual 
crystal, may be measured by determining the density of an inert liquid mixture 
in which the crystal remains just suspended. Examples of a convenient group of 
miscible organic liquids for many inorganic salts include chloroform 
(1492 kgm~ 3 at 20 °C), carbon tetrachloride (1594), ethyl iodide (1930), ethyl- 
ene dibromide (2180), bromoform (2890) and methylene iodide (3325). 

For example, a crystal of sodium nitrate (2260 kg m -3 ) could be floated in 
about 50 mL of bromoform in a suitable flask, taking care that no air bubbles 
are attached, and then caused to achieve the 'just suspended' state by slowly 
adding chloroform from a burette. At this point the crystal density may be 
assumed to be equal to that of the liquid mixture, which can readily be 

Solid densities have a very small temperature dependence, but this can be 
ignored for industrial crystallization purposes. For example the density of 
sodium chloride decreases by about 0.7 per cent when the temperature increases 
from 10 to 80 °C. The calculation needs a knowledge of the coefficient of 
thermal expansion. 

The densities of bulk particulate solids and slurries are discussed in section 

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Physical and thermal properties 33 

2.1.2 Liquids 

The density of a liquid is significantly temperature dependent. The ratio of the 
density of a given liquid at one temperature to the density of water at the same, 
or another, temperature is known as the specific gravity of the liquid. Thus, 
a specific gravity quoted at say 20 °C/4 °C is numerically equal to the density of 
the liquid at 20 °C expressed in gcm~ 3 since water exhibits its maximum density 
(lgcm- 3 )at4°C. 

The simplest instrument for measuring liquid density is the hydrometer, 
a float with a graduated stem. To approach reasonable accuracy, however, it 
is essential to make the measurement at the particular calibration temperature 
marked on the hydrometer. Densities may be determined more accurately by 
the specific gravity bottle method, or with a pyknometer (BS 733, 1983) or 
Westphal balance, details of which may be found in most textbooks of practical 

In recent years, several high-precision instruments have become available, 
the most noteworthy of which are those based on an oscillating sample holder. 
A glass U-tube is filled with the sample and caused to oscillate at its natural 
frequency, which is dependent on the total mass of the system. Since the tube 
has a constant mass and sample volume, the measured frequency of oscillation 
can be related to the liquid sample density. Precisions of up to ±10 _6 gcm~ 3 
have been claimed for some instruments. 

It is often possible to estimate to ±5% the density of a solution from a 
knowledge of the solute and solvent densities by means of the equation 

Psoln = Z" \ 1 - 1 ) 

— + — 

where L and S are the masses of the solvent (liquid) and solute (solid), 
respectively, and pl and ps are the densities of the respective components. It 
has to be acknowledged, of course, that the volume of a solution is not exactly 
equal to the volumes of the solvent and added solute, but the error incurred in 
making this assumption is often insignificant, particularly for industrial pur- 
poses, as the following examples show: 

1. An aqueous solution of potassium sulphate at 80°C containing 0.214kg 
K 2 S0 4 /kg water (p L = 971.8 kg m" 3 at 80 °C, p s = 2660 kg m" 3 ). 

p so i n = (1 + 0.214)/[(1/971.8) + (0.214/2660)] = 1095 kg m" 3 . 
Experimental value: 1117kgm~ 3 . 

2. An aqueous solution of sodium sulphate at 15°C containing 
0.429kg Na 2 S0 4 • 10H 2 O/kg 'free' water (p L = 999.1 kgnr 3 at 15°C, 
p s = 1460 kg m- 3 ). 

p soln = (1 + 0.429)/[(l/999.1) + (0.429/1460)] = 1103 kg m" 3 
Experimental value: 1125kgm~ 3 . 

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34 Crystallization 

3. An aqueous solution of sucrose at 80 °C containing 3.62 kg CgH^Oe/kg 
water (p L = 971.8kgm- 3 at 80°C, p s = 1590 kgrrr 3 ). 

Psoln = (1 + 3.62)/[(l/971.8) + (3.62/1590)] = 1397 kgrn" 3 
Experimental value: 1350kgm~ 3 . 

For more reliable estimations of solution density, reference should be made 
to the procedures described in the book by Sohnel and Novotny (1985) which 
also contains detailed density data for a large number of inorganic salt solu- 
tions. In a later publication (Novotny and Sohnel, 1988) densities of some 300 
inorganic salt solutions are recorded. 

The densities of some aqueous solutions are recorded in the Appendix 
(Table A. 9). 

2.1.3 Bulk solids and slurries 

The bulk density of a quantity of particulate solids is not a fixed property of the 
system since the bulk volume occupied contains significant amounts of void 
space, normally filled with air. The relationship between the density of the solid 
particles, p$, and the bulk solids density, p^s, is 

Pbs = Ps(l - e) (2.3) 

where e is the voidage, the volume fraction of voids, which is considerably 
dependent on particle shape, particle size distribution and the packing of the 
particles. Further, e can vary considerably depending on how the particulate 
material has been processed or handled. For example, it can be increased by 
aeration, as in freshly poured solids, and decreased by vibration, e.g. after 
transportation in packaged form. 

Expression of slurry densities 

Many different terms are used for specifying the solids content of slurries and 
each has its own particular use. 

Slurry concentrations are not usually simple to measure experimentally since 
it is not always convenient to filter-off, wash, dry and weigh the solids content. 
So other more convenient, but less precise methods, are often adopted. For 
example, it is common practice to take a sample of slurry in a graduated 
cylinder, allow the solids to settle and to measure the volume percentage of 
settled solids. Although this is often a rapid and quite satisfactory way of 
assessing the slurry concentration, particularly for routine testing under indus- 
trial plant conditions, it does not directly give the actual quantity of suspended 
solids, because the settled volume (the overall volume occupied by the settled 
solids) contains a significant proportion of liquid. Settled spheres of uniform 
size, for example, enclose a void space of about 40 per cent, but consider- 
able deviations from this value can occur for multisized particles of irregular 

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Physical and thermal properties 35 

The relationship between system voidages and the settled solids fraction is 
given by equation 2.4. Other useful relationships are given in equations 2.5 to 2.9. 

e = 1 - S{\ - e s ) (2.4) 

= 1 - (M T / PS ) (2.5) 

= (P- Ps)/(ps - Pl) (2.6) 

M T = (1 - e) Ps (2.7) 

= Xp (2.8) 

= X[ PS - e(p s - Ph )] (2.9) 

where S= settled solids fraction (m of settled solids plus associated liquor 

in the voids/m 3 of total sample taken) 
X=mass fraction of solids (kg of suspended solids/kg of total 
M t = slurry density (kg of suspended solids/m of total suspension) 
e = voidage of the slurry (m of liquid/m of total suspension) 
1 — e = volume fraction of solids (m of solids/m of total suspension) 
£s = voidage of the settled solids 
p = overall mean density of the suspension (kgm~ 3 ) 
ps = density of solid (kgm~ ) 
Pl = density of liquid (kgm~ ) 

For the special case of an industrial crystallizer, it is sometimes possible to 
assess the slurry (magma) density by chemical analysis, measuring (a) the total 
overall concentration of solute (crystals plus dissolved solute) in the suspension 
and (b) the concentration of dissolved solute in the supernatant liquor. Thus, 

C=M T + eC* (2.10) 


M T = PS (C - C*)/(ps " C*) (2.H) 

where C = kg of crystallizing substance (suspended and dissolved)/m 3 of total 
suspension, and C* = kg of dissolved crystallizing substance /m 3 of supernat- 
ant liquor. 

2.2 Viscosity 

The once common units of absolute viscosity, the poise (P) (1 gem -1 s~') and 
its useful sub-multiple the centipoise (cP), have now been replaced by the SI 
unit (kgm -1 s~') which is generally written as Pa s and sometimes as Nsm~ 2 . 
The following relationships hold: 

lcP = 0.01P= lmPas= lmNsnT 2 = lO^kgm^'s" 1 
= 2.421bfr 1 h- 1 


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36 Crystallization 

Similarly, the once common units of kinematic viscosity (= absolute viscosity/ 
density), the stokes (St) (lcm 2 s~') and its sub-multiple the centistokes (cSt), 
have been replaced by the SI unit (m 2 s ). The following relationships hold: 

lcSt= 0.01 St = 10 

- 6 mV 

0.0388 ft 2 IT 

The viscosity of a liquid decreases with increasing temperature, and for many 
liquids the relationship 

t] = A exp(-B/T) 


holds reasonably well. A and B are constants and the temperature T is 
expressed in kelvins. Plots of log 77 versus T or log 77 versus log T usually 
yield fairly straight lines and this property may be used for interpolating 
viscosities at temperatures within the range covered. 

In general, dissolved solids increase the viscosity of water, although a few 
exceptions to this rule are known. Occasionally, the increase in viscosity is 
considerable, as in the case of the system sucrose-water where, for example, the 
viscosity increases from around 2 to 60 mPa s for a concentration increase from 
20 to 60 g/100 g of solution at 20 °C. 

Figure 2.1a shows an example of a solute that decreases the viscosity of the 
solvent; in this system (Kl-water) a minimum viscosity is exhibited. Several 
other potassium and ammonium salts also exhibit a similar behaviour. Figure 
2.1b shows the effect of concentration and temperature on the ethanol-water 
system which exhibits a maximum viscosity. 

A comprehensive survey of the viscosity characteristics of aqueous solutions 
of electrolytes has been made by Stokes and Mills (1965) who also give experi- 
mental data on a considerable number of systems. Viscosities of some aqueous 
solutions are recorded in the Appendix (Table A. 10). 

Unfortunately, no completely reliable method is available for the prediction 
of the viscosities of solutions or liquid mixtures. A general survey is made by 


20 40 60 SO 

g of Kl/IOOml of solution 

'0 25 50 75 100 

g of ethanol/IOOg of solution 


Figure 2.1. Aqueous solutions exhibiting (a) minimum, (b) maximum viscosities. (After 
Hatschek, 1928) 

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Physical and thermal properties 



Figure 2.2. Simple U-tube viscometer 

Reid, Prausnitz and Poling (1987) and some indication of the complexities 
involved in making estimates of the viscosities of mixed salt solutions may be 
gained from the survey made by Nowlan, Thi and Sangster (1980). 

Numerous instruments have been devised for the measurement of liquid 
viscosity, many of which are based on the flow of a fluid through a capillary 
tube and the application of Poiseuille's law in the form 




where AP = the pressure drop across the capillary of length / and radius r, and 
V = volume of fluid flowing in unit time. One simple type of U-tube viscometer 
is shown in Figure 2.2. The liquid under test is sucked into leg B until the level in 
this leg reaches mark z. The tube is arranged truly vertical, and the temperature 
of the liquid is measured and kept constant. The liquid is then sucked up into 
leg A to a point above x and the time / for the meniscus to fall from x to y is 
recorded. The kinematic viscosity of the liquid v can be calculated from 



where k is a constant for the apparatus, determined by measurements on a 
liquid of known viscosity, e.g. water. 

The falling-sphere method of viscosity determination also has many applica- 
tions, and Stokes' law may be applied in the form 


(pg - P\)d 2 


where d, p and u are the diameter, density and terminal velocity, respectively, of 
a solid sphere falling in the liquid of density p\ . A simple falling-sphere visco- 
meter is shown in Figure 2.3. The liquid under test is contained in the inner 


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38 Crystallization 

Top of 

Small hole 


Figure 2.3. Falling-sphere viscometer 

tube of 30 mm diameter and about 300 mm length. The central portion of the 
tube contains two reference marks a and b, 1 50 mm apart. The tube is held truly 
vertical, and the temperature of the liquid is measured and kept constant. 
A 1.5 mm diameter steel ball, previously warmed to the test temperature, is 
inserted through a small guide tube and its time of passage between the two 
reference marks is measured. A mean of several measurements should be taken. 
The viscosity of the liquid can then be calculated from equation 2.15. Modern 
instrumental versions of the falling-sphere technique are claimed to measure 
viscosities in the range 0.5-500 mPa with high precision at controlled tempera- 
tures using sample volumes as low as 0.5 mL. 

Several high-precision viscometers are based on the concentric-cylinder 
method. The liquid under test is contained in the annulus between two vertical 
coaxial cylinders; one cylinder can be made to rotate at a constant speed, and 
the couple required to prevent the other cylinder rotating can be measured. For 
more detailed information on practical viscometry reference should be made to 
specialized publications (Dinsdale and Moore 1962, BS 188, 1993). 

2.2.1 Solid-liquid systems 

The viscosity characteristics of liquids can be altered considerably by the pres- 
ence of finely dispersed solid particles, especially of colloidal size. The viscosity 
of a suspension of rigid spherical particles in a liquid, when the distance 
between the spheres is much greater than their diameter, may be expressed by 
the Einstein equation: 

VS = ??o(l + 2.50) 


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Physical and thermal properties 39 

where 77s is the effective viscosity of the disperse system, 770 the viscosity of the 
pure dispersion medium, and <f> the ratio of the volume of the dispersed particles 
to the total volume of the disperse system. In other words, <j> = 1 — e, where e is 
the voidage of the system. Equation 2.16 applies reasonably well to lyophobic 
sols and very dilute suspensions, but for moderately concentrated suspensions 
and lyophilic sols the Guth-Simha modification is preferred: 

r/ s = 770(1 + 2.5^+14.1^) (2.17) 

For concentrated suspensions of solid particles, the Frankel-Acrivos (1967) 

(</# m ) 1/3 
8 ■'" \ (l - 4>/M l 

VS = oV«[ „ -rid (2-18) 

can be applied for values of 4>/4> m — > 1, where (f> m is the maximum attainable 
volumetric concentration of solids in the system (usually about 0.6 for packed 
monosize spherical particles). Equation 2.18 has met with experimental support 
in the region {4>j4> m ) > 0.7. However, for solids concentrations such as those 
normally encountered in industrial crystallizers (say e ~ 0.8, <j> ~ 0.2 and 
(t>l<pm ~ 0.3 for granular crystals) the much simpler equation 2.17 predicts the 
order of magnitude of apparent viscosity reasonably well. 

2.3 Surface tension 

Of the many methods available for measuring the surface tension of liquids 
(Findlay, 1973), the capillary rise and ring techniques are probably the most 
useful for general applications. 

In the capillary rise method, the surface tension, 7, of a liquid can be deter- 
mined from the height, h, of the liquid column in a capillary tube of radius r . 
If the liquid completely wets the tube (zero contact angle), 

j = \rhApg (2.19) 

where Ap is the difference in density between the liquid and the gaseous atmo- 
sphere above it. The height, h, can be accurately measured with a cathetometer 
from the base of the liquid meniscus to the flat surface of the free liquid surface 
in a containing vessel. However, to minimize errors, this reference to a flat 
surface can be eliminated by measuring the difference in capillary rise in two 
tubes of different bore {Figure 2.4). Then 

7 = \r x h\Apg = \r 2 h 2 Apg 

From which it follows that 

AhApr x r 2 g 
7 = ~7T r r- (2-20) 

2(ri - n) 

The differential height Ah can be measured with precision. 

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Narrow - bore capi I lary 

Wide-bore capillary 




Figure 2.4. Measurement of surface tension by the differential capillary rise method 

The ring technique, and its many variations, is widely used in industrial 
laboratories. Several kinds of commerical apparatus incorporating a torsion 
balance are available under the name du Noiiy tensometer. The method is 
simple and rapid, and is capable of measuring the surface tension of a pure 
liquid to a precision of 0.3% or better. 

The force necessary to pull a ring (usually of platinum or platinum-iridium 
wire) from the surface of the liquid is measured. The surface tension is calcu- 
lated from the pull and the dimensions of the ring after the appropriate 
correction factors have been applied. 

It is often possible to predict the surface tension of non-aqueous mixtures of 
solvents by assuming a linear dependence with mole fraction. Aqueous solu- 
tions, however, generally show a pronounced non-linear behaviour and predic- 
tion is not recommended. 

The surface tension of a liquid decreases with an increase in temperature, but 
the decrease is not always linear {Table 2.1). 

The addition of an electrolyte to water generally increases the surface tension 
very slightly, although an initial decrease is usually observed at very low 
concentrations (< 0.002 molL -1 ) (Harned and Owen, 1958). Non-electrolytes 
generally decrease the surface tension of water. For example, saturated aqueous 
solutions of a-naphthol, adipic acid and benzoic acid at 22 °C are 48, 55 and 
60mNm _1 , respectively, whereas a saturated solution of potassium sulphate at 
the same temperature has a surface tension of 73mNm _1 . 

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Physical and thermal properties 41 
Table 2.1. Surface tensions of some common solvents at different temperatures 


Surface tension 

, mN m ' 






50 °C 






















CC1 4 




























1 mNm ' = 1 mJm 2 = 1 dyncm 

2.3.1 Interfacial tension 

The surface tension of a liquid, as normally measured, is the interfacial tension 
between a liquid surface and air saturated with the relevant vapour. 

The interfacial tension of a crystalline solid in contact with a solution of the 
dissolved solid is a quantity of considerable importance in crystal nucleation 
and growth processes. It is also sometimes referred to as the 'surface energy'. 
This subject is dealt with in section 5.6. 

2.4 Diffusivity 

Two examples of a theoretical approach to the problem of the prediction of 
diffusion coefficients in fluid media are the equations postulated in 1905 by 
Einstein and in 1936 by Eyring. The former is based on kinetic theory and 
a modification of Stokes' law for the movement of a particle in a fluid, and is 
most conveniently expressed in the form 


D = — (2.21) 


where D = diffusivity (m 2 s _1 ), T = absolute temperature (K), r\ = viscosity 
(kgs^'m -1 ), r = molecular radius (m), k = Boltzmann's constant and the 
dimensionless factor <fi has a numerical value between 4vt and 6tt depending on 
the solute : solvent molecular size ratio. Eyring's approach, based on reaction 
rate theory, treats a liquid as a disordered lattice structure with vacant sites 
into which molecules move, i.e. diffuse. For low solute concentration Eyring's 
equation may be expressed in a form identical with that of equation 2.21 but 
a different value of <f> applies. 

The usefulness of these equations, however, is strictly limited because they 
both contain a term, r, which denotes the radius of the solute molecule. Values 
of this quantity are difficult to obtain. Consequently, the most directly useful 
relationship that emerges is 

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42 Crystallization 

Drj/T = constant (2.22) 

which is of considerable value in predicting, for a given system, the effect of 
temperature and viscosity on the diffusion coefficient. This simple relation is 
often referred to as the Stokes-Einstein equation. 

The very limited success of the theoretical equations has led to the develop- 
ment of many empirical and semi-empirical relationships for the prediction of 
diffusion coefficients. Amongst those devised for diffusion in liquids the fol- 
lowing may be mentioned. For diffusion in aqueous solutions Othmer and 
Thakar (1953) proposed the correlation 


" „l.l v 0.6 V-t-3) 

where v is the molar volume of the solute (cm 3 moP ) and r\ is the viscosity in 
cP, giving the diffusivity D in cm 2 s~'. By correlating a large number of 
published experimental diffusivities Wilke and Chang (1955) arrived at the 

D = 7.4xlO-' W o r (2.24) 

where M is the molar mass (kgkmol - ), v is the molal volume (cm 3 mol" ) and 
T is in kelvins, giving D in cm s _1 . For unassociated solvents, e.g. benzene, 
ether and heptane, the so-called association parameter 7=1. For water, 
methanol and ethanol, 7 = 2.6, 1.9 and 1.5, respectively. 

Despite the widespread use of these and many other similar correlations, 
however, they are notoriously unreliable; deviations from experimental values 
as high as 30% are not unusual (Mullin and Cook, 1965). Furthermore, these 
empirical relationships were devised from diffusion data predominantly on 
liquid-liquid systems, and there is little evidence to suggest that they are reliable 
for the prediction of the diffusion of solid solutes in liquid solutions, although 
an 'order of magnitude' estimation is sometimes possible. For example, the 
diffusivity of sodium chloride in water at 25 °C is 1.3 x 10~ 9 m 2 s~', while 
values calculated from equations 2.23 and 2.24 range from 1.7 to 2.6 x 10~ 9 
m 2 s _1 . Similarly, for sucrose in water at 25 °C, D = 5.2 x 10 _10 m 2 s _1 , while the 
predicted values range from 3.6 to 4.2 x 10~'°m 2 s _1 . 

It is also important to note that these empirical correlations are meant to 
apply only to dilute solutions. Despite the fact that they all contain a term 
relating to viscosity which is a function of concentration, they usually fail to 
predict the rate of diffusion from a concentrated solution to a less concentrated 
one. For example, the diffusion coefficient for sucrose diffusing from a 1% 
aqueous solution into water at 25 °C is approximately five times the value for 
the diffusion between 61.5 and 60.5% solutions, whereas over the concentra- 
tion range 1 to 60%, the viscosity exhibits a fortyfold increase. 

These empirical equations also fail to discriminate between isomers as 
was pointed out by Mullin and Cook (1965), who measured the diffusivities 
of 0-, m- and />-hydroxybenzoic acid in water. The data measured at 20 °C are 

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Physical and thermal properties 



Temperature, °C 

Figure 2.5. Diffusivities of saturated aqueous solutions of the hydroxybenzoic acids into 

water: O = ortho-, □ = meta-, A = para-, = predicted from equation 2.24 (after 

Mullin and Cook, 1965) 

compared with values predicted by equation 2.24 in Figure 2.5, where the 
deviation is about ±30% between the estimated values and those measured 
for the m- and/?-isomers, with a difference of about 60% between the o- and m-. 
Clearly equation 2.24 and other empirical relationships, fail to take some 
property of the system into account, and it is likely that this quantity is the 'size' 
of the diffusing component. For the case of the hydroxybenzoic acids the 
differences in diffusivity can be accounted for by considering the different 
hydrogen bonding tendencies of the three isomers, which in turn would influ- 
ence both the size and shape of the diffusing species. 

2.4.1 Experimental measurements 

In a diffusion cell, where two liquids are brought into contact at a sharp 
boundary, three different states of diffusion may be recognized. In the case of 
'free' diffusion, concentrations change progressively away from the interface; 
when concentrations begin to change at the ends of the cell, 'restricted' diffu- 
sion is said to occur. If the concentration at a given point in the cell remains 
constant with respect to time, 'steady-state' diffusion is taking place, and, as in 
all other steady-state processes, a constant supply of material to and removal 
from the system is required. Several comprehensive accounts have been given of 
the methods used for measuring diffusion coefficients under these three condi- 
tions (Tyrrell, 1961; Stokes and Mills, 1965; Robinson and Stokes, 1970). 

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44 Crystallization 

The techniques used for restricted and steady-state diffusion generally 
involve the use of a diaphragm cell. This method has the disadvantage that 
the cell has first to be calibrated with a system with a known diffusion coeffi- 
cient, and for systems with relatively slow diffusivities each run may require an 
inconveniently long time. 

The diffusion coefficient under free-diffusion conditions can be measured by 
analysis at the termination of the experiment or by continuous or intermittent 
analysis while diffusion continues. Care has to be taken not to disturb the 
system, and the most widely employed methods for measuring diffusion coeffi- 
cients in liquids are interferometric, resulting from the original work of Gouy in 
the 19th century. 

Two kinds of diffusivity can be recorded, viz. the differential, D, and the 
integral, D. The differential diffusivity is a value for one particular concentra- 
tion, c, and driving force, c\ — cj, where c = (c\ + C2)/2 and c\ — a is suffi- 
ciently small for D to remain unchanged over the concentration range. 
However, the diffusion coefficient is usually concentration dependent, and in 
most cases it is the integral diffusivity that is normally measured. This is an 
average value over the concentration range c\ to ci. 

The differential diffusivity is of considerable theoretical importance, and it is 
only through this quantity that experimental measurements by different tech- 
niques can be compared. On the other hand, it is the integral coefficient that is 
generally required for mass transfer assessment, since this coefficient represents 
the true 'average' diffusivity over the concentration range involved in the mass 
transfer process. 

For example, in the dissolution of a solid into a liquid, the solute diffuses 
from the saturated solution at the interface to the bulk solution. In crystal- 
lization the solute diffuses from the supersaturated bulk solution to the satur- 
ated solution at the interface. The relevant diffusivity that should be used in an 
analysis of these two processes, therefore, is the integral diffusivity, which 
covers the range of concentration from equilibrium saturation to that in the 
bulk solution. 

Integral diffusivities may be measured directly by the diaphragm cell tech- 
nique (Dullien and Shemilt, 1961), but unless the concentration on both sides of 
the diaphragm (see Figure 2.6) is maintained constant throughout the experi- 
ment, the diffusivity measured is the rather complex double-average known as 
the 'diaphragm cell integral diffusivity'. D&, defined by an integrated form of 
Fick's law of diffusion: 

D d = Un( C -^^) (2.25) 

where Co and c t are the initial and final (at time t) concentrations, 1 and 2 refer 
to the lower and upper cells, and (3 is the cell constant (m~ 2 ), which can be 
calculated from the cell dimensions: 

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Physical and thermal properties 45 

Hp;l G 

Figure 2.6. Diaphragm diffusion cell (after Dullien and Shemilt, 1961). A, light liquid 
compartment; B, heavy liquid compartment; C, stop-cock; D, E, capillaries; F, sintered 
glass diaphragm; G = polythene-coated iron stirrers; H, rotating magnets 

where A and L = area and thickness of diaphragm, and V\ and V2 = volume of 
the cell compartments. 

The above method is extremely time-consuming and the process of convert- 
ing integral to differential values is both tedious and inaccurate. It is more 
practicable, therefore, to measure the differential diffusivity, D, at intervals 
over the whole concentration range and to calculate the required integral 
diffusivity, D, by means of the relationship 



C\ - c 2 



Figure 2.7 shows the measured differential and calculated integral diffusiv- 
ities for the systems KC1— H2O and NH4CI— H2O (Nienow, Unahabhokha and 

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46 Crystallization 

1.8 - 



■ — 














NH 4 Ct 



O.l 0.2 0.3 

Concentration, g/g of H,0 


Figure 2.7. Measured differential (•) awe? calculated integral {broken line) diffusivities for 
the systems NH4CI— H2O and KCl— H2O at 20°C (after Nienow, Unahabhokha and 
Mullin, 1968) 

Mullin, 1968). Diffusivities for a number of common electrolyte systems are 
given in the Appendix [Table A. 11). 

The diffusivity of a strong electrolyte at infinite dilution is called the Nernst 
limiting value of the diffusion coefficient, D°, which can be calculated from 

D° = R7 >' " Vl) ■ AoiA ° 2 (2.28) 

F 2 ^kil A01 + A02 

where A01 and A02 are the limiting conductivities, and v\ and vj are the number 
of cations and anions of valency z\ and z^, respectively. Using the conditions of 

v\Z\ + V2Z2 = 

so equation 2.28 may also be written 

RT \zi\ + \z 2 \ A01A02 


\z\z 2 \ 





where F is the Faraday constant (9.6487 x 10 4 Cmol ), R is the gas constant 

(8.3143 JKT'mor 1 ), giving values of RT/¥ Z of 2.4381, 2.6166 and 
2.7951 x 10~ 2 S~' mol -1 s _1 at 0, 20 and 40 °C, respectively. These may be used 
with the values of An (S m 2 mol~ ) in the Appendix (Table A. 13) to give values 
of D° in m 2 s . 

By the application of reaction rate theory to both viscosity and diffusion it 
can be shown that 

r\ = A exp(— Ey/RT) 


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Physical and thermal properties 



D= Bexp(-E D /RT) 


A plot of log ry against \jT should yield a straight line, and similarly a linear 
relationship should exist between logZ) and \/T. The slopes of these plots give 
the respective energies of activation Ey (viscosity) and Eq (diffusion). 

2.5 Refractive index 

Refractometric measurements can often be used for the rapid measurement of 
solution concentration. Several standard instruments (Abbe, Pulfrich, etc.) are 
available commercially. A sodium lamp source is most usually used for illu- 
mination, and an instrument reading to the fourth decimal place is normally 
adequate for crystallization work. It is advisable that calibration curves be 
measured, in terms of temperature and concentration, prior to the study with 
the actual system. 

If a dipping-type refractometer is used, a semi-continuous measurement 
may be made of the change in concentration as the system crystallizes. 
However, if nucleation is heavy or if large numbers of crystals are present, 
it may be difficult to provide sufficient illumination for the prism because of 
the light scattering. One solution to this problem (Leci and Mullin, 1968) is 
to use a fibre optic (a light wire) fitted into a collar around the prism 
illuminated from an external source (Figure 2.8). In this way undue heating 
of the solution is also avoided. 


Figure 2.8. A technique for illuminating the prism of a dipping-type refractometer in an 
opaque solution: A, Perspex collar; B, fibre optic holder; C, fibre optic; D, refractometer 
prism; E, polished face of prism 

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2.6 Electrolytic conductivity 

The electrolytic conductance of dilute aqueous solutions can often be measured 
with high precision and thus afford a useful means of determining concentra- 
tion (see section 3.6.3). A detailed account of the methods used in this area is 
given by Robinson and Stokes (1970). 

In the author's experience, however, conductivity measurements are of 
limited use in crystallization work because of the unreliability of measurement 
in near-saturated or supersaturated solutions. The temperature dependence of 
electrical conductivity usually demands a very high precision of temperature 
control. Torgesen and Horton (1963) successfully operated conductance cells 
for the control of ADP crystallization, but they had to control the temperature 
to ±0.002 °C. 

2.7 Crystal hardness 

Crystals vary in hardness not only from substance to substance but also from 
face to face on a given crystal (Brookes, O'Neill and Redfern, 1971). One of the 
standard tests for hardness in non-metallic compounds and minerals is the 
scratch test, which gave rise to the Mohs scale. Ten 'degrees' of hardness are 
designated by common minerals in such an order that a given mineral will 
scratch the surface of any of the preceding members of the scale (see Table 2.2). 
The hardness of metals is generally expressed in terms of their resistance to 
indentation. A hard indenter is pressed into the surface under the influence of 
a known load and the size of the resulting indentation is measured. A widely 
used instrument is the Vickers indenter, which gives a pyramidal indentation, 
and the results are expressed as a Vickers hardness number (kgf mm~ ). Other 

Table 2.2. Mohs scale of hardness 














3MgO • 4Si0 2 • H 2 




CaSC-4 • 2H 2 




CaC0 3 




CaF 2 




CaF 2 • 3Ca 3 (P0 4 ) 2 




K 2 • A1 2 3 • 6Si0 2 




Si0 2 




(A1F) 2 • Si0 4 




A1 2 3 

2 000 





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Physical and thermal properties 49 

tests include the Rockwell, which uses a conical indenter and the Brinell, which 
uses a hard steel ball (Bowden and Tabor, 1964). 

The relation between Mohs hardness, M, and the Vickers hardness, V, is not 
a clear one. However, if diamond (M = 10) is omitted from the Mohs scale, the 

\ogV = 0.2M + 1.5 (2.32) 

may be used for rough approximation purposes for values of M < 9. 

A few typical surface hardnesses (Mohs) of some common substances are: 

















The scratch test is not really suitable for specifying the hardness of 
substances commonly crystallized from aqueous solutions, because their Mohs 
values lie in a very short range, frequently between 1 and 3 for inorganic salts 
and below 1 for organic substances. For a reliable measurement of hardness of 
these soft crystals the indentation test is preferred. Ridgway (1970) has meas- 
ured mean values of the Vickers hardness for several crystalline substances: 

sodium thiosulphate (Na2S2C>3 • 5H2O) 

potassium alum (KA1(S0 4 ) 2 • 12H 2 0) 

ammonium alum (NH 4 A1(S0 4 ) 2 • 12H 2 0) 

potassium dihydrogen phosphate (KH 2 P04) 


sucrose (C12H22O11) 

He has also determined the hardnesses of different faces of the same crystal: 

ammonium dihydrogen phosphate (NH4H 2 P04) (100) 


potassium sulphate (K2SO4) (100) 







Hardness appears to be closely related to density (proportional to) and to 
atomic or molecular volume (inversely proportional), but few reliable data 
are available. In a recent study, Ulrich and Kruse (1989) made some interesting 
comments on these relationships and confirmed the need for more experimental 
data before any acceptable prediction method can be developed. 

2.8 Units of heat 

The SI heat energy unit is the joule (J), but four other units still commonly 
encountered are the calorie (cal), kilocalorie (kcal), British thermal unit (Btu) 
and the centigrade heat unit (chu). The old definitions of these four units are 

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50 Crystallization 

Table 2.3. Equivalent values of the common heat energy units 








0.003 97 

0.002 21 













0.453 6 




0.238 8 

2.388 x 10~ 4 

9.478 x 10~ 4 

5.275 x 10~ 4 


based on the heat energy required to raise the temperature of a unit mass of 
water by one degree: 

1 cal = 1 g of water raised through 1 °C (or K) 
1 kcal= 1 kg of water raised through 1 °C (or K) 
1 chu = 1 lb of water raised through 1 °C (or K) 
1 Btu = 1 lb of water raised through 1 °F 

The definitions of these units are linked to the basic SI unit, the joule: 

1J= lWs= INm 

Table 2.3 indicates the equivalent values of these various heat units. 

2.9 Heat capacity 

The amount of heat energy associated with a given temperature change in 
a given system is a function of the chemical and physical states of the system. 
A measure of this heat energy can be quantified in terms of the quantity known 
as the heat capacity which may be expressed on a mass or molar basis. The 
former is designated the specific heat capacity (Jkg -1 K ) and the latter the 
molar heat capacity (Jmol~' K~'). The relationships between some commonly 
used heat capacity units are: 

specific heat capacity, C 

1 cal g _1 °C _1 (or KT 1 ) = 1 Btu lb" 1 F~' 

= lchulb _lo C _1 (orr 1 ) 
= 4.187kJkg~ 1 KT l 
molar heat capacity, C 

1 calmoF 1 °C (or KT 1 ) = 1 Btulb-moP 1 "F -1 

= 1 chulb-moF 1 °C' (or KT 1 ) 
= 4.187Jmor 1 KT l 

For gases two heat capacities have to be considered, at constant pressure, C p , 
and at constant volume, C v . The value of the ratio of these two quantities, 
C p /C v = 7, varies from about 1.67 for monatomic gases (e.g. He) to about 1.3 

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Physical and thermal properties 51 

for triatomic gases (e.g. CO2). For liquids and solids there is little difference 
between C p and C v , i.e. 7 ~ 1, and it is usual to find C p values only quoted in 
the literature. 

2.9.1 Solids 

Specific heat capacities of solid substances near normal atmospheric temper- 
ature can be estimated with a reasonable degree of accuracy by combining two 
empirical rules. 

The first of these, due to Dulong and Petit, expresses a term called the 
'atomic heat' which is defined as the product of the relative atomic mass and 
the specific heat capacity. For all solid elemental substances, the atomic heat is 
assumed to be roughly constant: 

atomic heat ~ 6.2calmoP °C 

which in SI units is equivalent to approximately 26Jmol~ 1 K _1 . However, since 
virtually all the data available in the literature are recorded in calorie units, 
these will be retained in this section for all the examples. 

The second rule, due to Kopp, applies to solid compounds and may be 
expressed by 

molar heat capacity = sum of the atomic heats of the constituent atoms 

In applying these rules, the following exceptions to the approximation 'atomic 
heat ~ 6.2' must be noted: 


C= 1.8 

H = 2.3 

B = 2.7 


= 4.0 

F= 5.0 

S= 5.4 

[H 2 0] 

The substance [H2O] refers to water as ice or as water of crystallization in solid 
substances. Obviously a reliable measured value of a heat capacity is preferable 
to an estimated value, but in the absence of measured values, Kopp's rule can 
prove extremely useful. A few calculated and observed values of the molar heat 
capacity are compared in Table 2.4. 

Table 2.4. Estimated (Kopp's rule) and observed values of molar heat capacity of several 
solid substances at room temperature 





u c -i 

Sodium chloride 


6.2 + 6.2 



Magnesium sulphate 

MgS0 4 • 7H 2 

6.2 + 5.4 + 4(4.0) + 7(9.8) 




C 6 H 5 I 

6(1.8) + 5(2.3) + 6.2 





10(1.8) + 8(2.3) 



Potassium sulphate 


2(6.2) + 5.4 + 4(4.0) 



Oxalic acid 

C 2 H 2 4 • 2H z O 

2(1.8) + 2(2.3) + 4(4.0) 

+ 2(9.8) 



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52 Crystallization 

Many inorganic solids have values of specific heat capacity, c p , in the range 
O.l-O.Scalg^CT'CO^-lJkJkmor'K- 1 ) and many organic solids have 
values in the range 0.2-0.5 (0.8-2.1). In general, the heat capacity increases 
slightly with an increase in temperature. For example, the values of c p at 
and 100 °C for sodium chloride are 0.21 and 0.22 calg -1 °C~' (8.8 and 
9.2kJkmol~ 1 K ) respectively, and the corresponding values for anthracene 
are 0.30 and 0.35 (approximately 13 and 15kJkmol _1 K _1 ). 

2.9.2 Pure liquids 

A useful method for estimating the molar heat capacity of an organic liquid is 
based on the additivity of the heat capacity contributions [C] of the various 
atomic groupings in the molecules (Johnson and Huang, 1955). Table 2.5 lists 
some [C] values, and the following examples illustrate the use of the method - 
the molar capacity values (calmoC °C ) in parentheses denote values 
obtained experimentally at 20 °C: 

methyl alcohol (CH 3 • OH) 9.9 + 11.0 = 20.9(19.5) 
toluene (C 6 H 5 • CH 3 ) 30.5 + 9.9 = 40.4 (36.8) 

CH 3 

isobutyl acetate CH 3 • COO • CH 

CH 2 • CH 3 

= 3(9.9) + 14.5 + 5.4 + 6.3 = 55.9 (53.3) 

The heat capacity of a substance in the liquid state is generally higher than that 
of a substance in the solid state. A large number of organic liquids have specific 
heat capacity c values in the range 0.4-0.6 calg -1 "C 1 (1.7-2.5 kJkg -1 K _1 ) at 
about room temperature. The heat capacity of a liquid usually increases with 
increasing temperature: for example, the values of c for ethyl alcohol at 0, 20 

Table 2.5. Contributions of various atomic groups to the molar heat 
capacity (calmoF C C _1 ) of org 
Johnson and C. J. Huang, 1955) 

capacity (calmol ' °C ) of organic liquids at 20 °C (after A. I. 





C6H5 — 




CH 3 — 


— NO, 


— CH 2 — 


— NH 2 


— CH 


— CN 






— COO— 



— Br 


C=0 (ketones) 


— S— 


— H (formates) 


— O — (ethers) 


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Physical and thermal properties 53 

and 60 °C are 0.54, 0.56 and 0.68 calg" 1 °C' (2.3, 2.3 and 2.9kJkg-' K" 1 ) and 
those for benzene at 20, 40 and 60 °C are 0.40, 0.42 and 0.45 (1.7, 1.8 and 
2.1 kJkg -1 K~'). Water is an exceptional case; it has a very high heat capacity 
and exhibits a minimum value at 30 °C. The values for water at 0, 15, 30 and 
100 °C are 1.008, 1.000, 0.9987 and 1.007calg- 1 °C _1 , respectively. 

2.9.3 Liquid mixtures and solutions 

Although not entirely reliable, the following relationship may be used to predict 
the molar heat capacity C of a mixture of two or more liquids: 

Cmixt = x A C A + x B C B H (2.33) 

where x denotes the mole fraction of the given component in the mixture. 
A similar relationship may be used to give a rough estimate of the specific heat 
capacity c: 

Cmixt = ^At'A + ^BCB H (2.34) 

where X is a mass fraction. 

For example, the value of c for methanol at 20 °C is 2.4kJkg~' K '. From 
equation 2.34 it can be calculated that an aqueous solution containing 75 mass 
per cent of methanol has a specific heat capacity of 2.9kJkg~' K , which 
coincides closely with measured values. 

For dilute aqueous solutions of inorganic salts, a rough estimate of the 
specific heat capacity can be made by ignoring the heat capacity contribution 
of the dissolved substance, i.e. 

|c| = |l-y| (2.35) 

where Y = mass of solute/mass of water, and c = calg -1 °C~ . Thus, solutions 
containing 5g NaCl, 10 g KC1 and 15 g CUSO4 per 100 g of solution would by 
this method be estimated to have specific heat capacities of 0.95, 0.89 and 
0.82calg~ lo C~', respectively. Measured values (25 °C) for these solutions are 
0.94, 0.91 and 0.83calg- lo C _1 (3.9, 3.8 and 3.5kJkg-' K" 1 ), respectively. This 
estimation method cannot be applied to aqueous solutions of non-electrolytes 
or acids. 

Another rough estimation method for the specific heat capacity of aqueous 
solutions is based on the empirical relationship 

|c| = |P _1 | (2-36) 

where p = density of the solution in gcm~ 3 . For example, at 30 °C a 2 per cent 
aqueous solution of sodium carbonate by mass has a density of 1.016 gem -1 
and a specific heat capacity of 0.98 calg -1 °C~ (1/p = 0.98), while a 
20 per cent solution has a density of 1.210 and a specific heat capacity of 0.86 
(l/p = 0.83). 

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54 Crystallization 

2.9.4 Experimental measurement 

The specific heat capacity of a liquid may be measured by comparing its cooling 
rate with that of water. This is most conveniently done in a calorimeter, 
a copper vessel fitted with a copper lid and heavy copper wire stirrer, supported 
in a draught-free space kept at constant temperature. The vessel is filled almost 
to the top with a known mass of water at about 70 °C and its temperature is 
recorded every 2 minutes as it cools to, say, 30 °C. Significant errors may be 
incurred if the starting temperature exceeds about 70 °C because evaporation 
greatly increases the rate of cooling. The water is then replaced by the same 
volume of the liquid or solution under test and its cooling curve is determined 
over the same temperature range. 

Taking the specific heat capacities of copper and water to be 385 and 4185 
Jkg^ 1 K ' respectively, a balance of the heat losses gives 

(4185m w + 385m c )(d(9/d/) w = (dim + 385m c )(d0/dOi 

where m c = mass of the copper calorimeter, lid and stirrer, m w = mass of water 
and m\ = mass of liquid. The cooling rates (d9/dt) are taken from the slopes of 
the cooling curves at the chosen temperature 0, and the specific heat capacity c\ 
of the liquid under test can thus be evaluated. 

2.10 Thermal conductivity 

The thermal conductivity, k, of a substance is defined as the rate of heat 
transfer by conduction across a unit area, through a layer of unit thickness, 
under the influence of a unit temperature difference, the direction of heat 
transmission being normal to the reference area. Fourier's equation for steady 
conduction may be written as 

%, = -< 

where q, t, A, 9 and x are units of heat, time, area, temperature and length 
(thickness), respectively. 

The SI unit for thermal conductivity is Wm~'K~' although other units such 
as cal s _1 cm _lo C _1 and Btu h _1 ft _1 °F _1 are still commonly encountered. The 
conversion factors are: 

1 cals" 1 cirT 1 °C' = 418.7WirT 1 K" 1 = 241.9 Btuh" 1 ft -1 F _1 

The thermal conductivity of a crystalline solid can vary considerably accord- 
ing to the crystallographic direction, but very few directional values are avail- 
able in the literature. Some overall values (Wm~' K~') for polycrystalline or 
non-crystalline substances include KC1 (9.0), NaCl (7.5), KBr (3.8), NaBr (2.5), 
MgS0 4 • 7H 2 0(2.5), ice (2.2),K 2 Cr 2 7 (1.9), borosilicate glass (1.0), soda glass 
(0.7) and chalk (0.7). 

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Physical and thermal properties 55 
Table 2.6. Thermal conductivities of some pure liquids 


Thermal conductivity, 







Benzene Methanol 


CC1 4 






















An increase in the temperature of a liquid usually results in a slight decrease 
in thermal conductivity, but water is a notable exception to this generaliza- 
tion. Furthermore, water has a particularly high thermal conductivity 
compared with other pure liquids {Table 2.6). There are relatively few values 
of thermal conductivity for solutions recorded in the literature and regrett- 
ably they are frequently conflicting. There is no generally reliable method of 

The thermal conductivity of an aqueous solution of a salt is generally slightly 
lower than that of pure water at the same temperature. For example, the values 
at 25 °C for water, and saturated solutions of sodium chloride and calcium 
chloride are 0.60, 0.57 and 0.54Wm~' K ', respectively. 

The thermal conductivity K(Wm~' K _1 ) of pure liquids between about and 
70 °C may be roughly estimated by the equation 

k = 3.6 x 10 _8 cp(p/M) 1/3 (2.38) 

where c is the specific heat capacity (Jkg^'K -1 ), p is the density (kgm~ 3 ) and 
M is the molar mass (kgkmol -1 ). 

2.11 Boiling, freezing and melting points 

When a non-volatile solute is dissolved in a solvent, the vapour pressure of the 
solvent is lowered. Consequently, at any given pressure, the boiling point of 
a solution is higher and the freezing point lower than those of the pure solvent. 
For dilute ideal solutions, i.e. such as obey Raoult's law, the boiling point 
elevation and freezing point depression can be calculated by an equation of 
the form 

AT = (2.39) 

where m = mass of solute dissolved in a given mass of pure solvent and M = 
molar mass of the solute. When AT" refers to the freezing point depression, 
K = K(, the cryoscopic constant; when AT refers to the boiling point elevation, 
K = Kb, the ebullioscopic constant. Values of K[ and K\, for several common 
solvents are given in Table 2.7; these, in effect, give the depression in freezing 
point, or elevation in boiling point, in °C when 1 mol of solute is dissolved, 
without dissociation or association, in 1 kg of solvent. 

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56 Crystallization 

Table 2.7. Cryoscopic and ebutlioscopic constants for some common solvents 




K { 

K b 




Kkgmol -1 



Acetic acid 




















Carbon disulphide 





Carbon tetrachloride 




















Methyl alcohol 















The cryoscopic and ebullioscopic constants can be calculated from values of 
the enthalpies of fusion and vaporization, respectively, by the equation 

R T^ 
K =AH (2M) 

When K = K[, T refers to the freezing point T[ (K) and AH to the enthalpy of 
fusion, A//f(Jkg~ ). When K=K\ ) ,T refers to the boiling point T^ and 
A// = A// V b, the enthalpy of vaporization at the boiling point. The gas con- 
stant R = 8.314 J mor'K- 1 . 

Boiling points and freezing points are both frequently used as criteria for the 
estimation of the purity of near-pure liquids. Detailed specifications of stand- 
ard methods for their determination are given, for example, in the British 
Pharmacopoeia (2000). 

Equation 2.40 cannot be applied to concentrated solutions or to aqueous 
solutions of electrolytes. In these cases the freezing point depression cannot 
readily be estimated. The boiling point elevation, however, can be predicted 
with a reasonable degree of accuracy by means of the empirical Diihring rule: 
the boiling point of a solution is a linear function of the boiling point of the 
pure solvent. Therefore, if the boiling points of solutions of different concen- 
trations are plotted against those of the solvent at different pressures, a family 
of straight lines (not necessarily parallel) will be obtained. A typical Diihring 
plot, for aqueous solutions of sodium hydroxide, is given in Figure 2.9 from 
which it can be estimated, for example, that at a pressure at which water boils at 
80 °C, a solution containing 50 per cent by mass of NaOH would boil at about 
120 °C, i.e. a boiling point elevation of about 40 °C. 

The boiling point elevation (BPE) of some 40 saturated aqueous solutions 
of inorganic salts have been reported by Meranda and Furter (1977) who 
proposed the correlating relationship 

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Physical and thermal properties 






O 1 




20 40 60 80 100 

Boiling point of water, °C 

Figure 2.9. DiXhring plot for aqueous solutions of sodium hydroxide 

BPE(°C)= 104.9x 



where Xs is the mole fraction of salt in the solution. 

The practical difficulties of measuring boiling point elevations have been 
discussed by Nicol (1969). 

2.11.1 Melting points 

The melting point of a solid organic substance is frequently adopted as a 
criterion of purity, but before any reliance can be placed on the test, it is 
necessary for the experimental procedure to be standardized. Several types of 
melting point apparatus are available commercially, but the most widely used 
method consists of heating a powdered sample of the material in a glass 
capillary tube located close to the bulb of a thermometer in an agitated bath 
of liquid. 

The best type of glass tube is about 1 mm internal diameter, about 70 mm 
long, with walls about 0.1 mm thick. The tube is heat-sealed at one end, and the 
powdered sample is scraped into the tube and knocked or vibrated down to the 
closed end to give a compacted layer about 5 mm deep. It is then inserted into 
the liquid bath at about 10 °C below the expected melting point and attached 
close to the thermometer bulb. The bath is agitated and temperature is raised 
steadily at about 3 °C/min. The melting point of the substance is taken as the 
temperature at which a definite meniscus is formed in the tube. For pure 
substances the melting point can be readily and accurately reproduced; for 
impure substances it is better to record a melting range of temperature. 

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58 Crystallization 

Table 2.8. Melting points of pure organic compounds 
useful for calibrating thermometers 


Melting point, 


Phenyl salicylate (salol) 










Benzoic acid 




Salicylic acid 


Succinic acid 




/>-Nitrobenzene acid 




Some organic substances are extremely sensitive to the presence of traces of 
alkali in soft soda-glass capillary tubes, giving unduly low melting points. In 
these cases, borosilicate glass tubes are recommended. Traces of moisture in the 
tube will also lower the melting point so the capillary tubes should always be 
stored in a desiccator. 

Some organic substances begin to decompose near their melting point, so 
that it is important not to keep the sample at an elevated temperature for 
prolonged periods. The insertion of the capillary into the bath at 10 °C below 
the melting point, allowing the temperature to rise at 3 °C/min, usually elim- 
inates any difficulties. For reproducible results to be obtained, the sample 
should be in a finely divided state (<100um). The thermometer, preferably 
graduated in increments of 0.1 °C, should be accurately calibrated over its 
whole range and, if in constant use, should be checked regularly. The well- 
known standardization temperature are the freezing and boiling points of 
water, but other standards that can be used are the melting points of pure 
organic substances, such as those indicated in Table 2.8. 

The liquid used in the heating bath depends on the working temperature; 
water is quite suitable for melting points from about room temperature to 
about 70 °C and liquid paraffin and a range of silicone fluids are widely used 
for more elevated temperatures. 

Specifications of methods for determining melting points are given in the 
British Pharmacopoeia (2000). See also section 4.5.1. 

2.12 Enthalpies of phase change 

When a substance undergoes a phase change, a quantity of heat is transferred 
between the substance and its surrounding medium. Several types of enthalpy 

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Physical and thermal properties 


change, still often referred to as the latent heat, and the following types may be 

solid I ^ solid II 
solid ^ liquid 
solid ^ gas 

liquid 5=± gas 

(enthalpy of transition, A// t ) 
(enthalpy of fusion, A//f) 
(enthalpy of sublimation, AH S ) 
(enthalpy of vaporization, A// v ) 

Only the last three of these represent significant quantities of heat energy: 
enthalpies of transition can for most industrial purposes be ignored. For 
example, the transformation of monoclinic to orthorhombic sulphur is accom- 
panied by an enthalpy change of about 2 kJ kg" , whereas the fusion of ortho- 
rhombic sulphur is accompanied by an enthalpy change of about 70 kJ kg" . 

Enthalpy changes, like the other thermal properties, can be expressed on a 
mass or molar basis, but to avoid confusion, all in this section will be expressed 
on a molar basis. The relationship between some commonly used units are: 


lcalg" 1 = lchulb" 1 = 1.8Btulb 


1 calmol 

1 chu lb-mol 

4.187kJkg" 1 
1.8 Btu lb-mol" 1 =4.187Jmor 1 

The relationship between any enthalpy change AH and the pressure- 
volume-temperature conditions of a system is given by the Clapeyron equation 





where dp/dT = rate of change of vapour pressure with absolute temperature 
and Av = volume change accompanying the phase change. 

A typical temperature-pressure phase diagram for a one-component system 
is shown in Figure 2.10. The sublimation curve AX indicates the increase of the 
vapour pressure of the solid with an increase in temperature. This is expressed 
quantitatively by the Clapeyron equation written as 

line \ 




X Vaporization 



Figure 2.10. Temperature-pressure diagram for a single-component system 

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60 Crystallization 
'dp\ AH S 

dTj s T(v g - v s ) 


where v 2 and v s = the molar volumes of the vapour and solid, respectively, and 
A// s = enthalpy of sublimation. The vaporization curve XB indicates the 
increase of the vapour pressure of the liquid with an increase in temperature: 

dp\_ A// v 

d7y v T(vg-vi) 


where Vi = the molar volume of the liquid, and AH V = latent heat of vaporiza- 
tion. Curve XB is not a continuation of curve AX; this fact can be confirmed by 
calculating their slopes dp/dT from equations 2.43 and 2.44 at point X. 

The fusion line X C indicates the effect of pressure on the melting point of the 
solid; it can either increase or decrease with an increase in pressure, but 
the effect is so small that line XC deviates only slightly from the vertical. When 
the fusion line deviates to the right, as in Figure 2.10, the melting point increases 
with an increase in pressure, and the substance contracts on freezing. Most 
substances behave in this manner. When the line deviates to the left, the melting 
point decreases with increasing pressure, and the substance expands on freez- 
ing. Water (see section 4.4) and type metals (Pb-Bi alloys) are among the few 
examples of this behaviour that can be quoted. The equation for the fusion 
line is 

d7y t 7\vi - v s ) 

where AH[ is the enthalpy of fusion. 

The enthalpies of sublimation, vaporization and fusion are related by 

AH S = AH f + AH V (2.46) 

but this additivity is applicable only at one specific temperature. The variation 
of an enthalpy change with temperature can be calculated from the Clausius 


-, Trr = c 2 - ci (2.47) 

dT T 

When AH = AH V , c\ and a are the molar heat capacities of the liquid, just on 
the point of vaporization, and of the saturated vapour, respectively. Equation 
2.47 can also be used for calculating A//f and AH S the appropriate values of c 
being inserted. 

The specific volumes v\ and v s are much smaller than v g ; equations 2.43 and 
2.44 can therefore be simplified to 

dp AH 

-£= = (2.48) 

dT Tv B 

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Physical and thermal properties 61 

where AH = AH V or A// s . From the ideal gas laws v g = RT/p, so that equa- 
tion 2.48 may be written 

This is the Clausius-Clapeyron equation. If the enthalpy change is considered 
to be constant over a small temperature range, T\ to T 2 , equation 2.49 may be 
integrated to give 

lnPl= AH(T 2 -T l) 
Pi RTiT 2 

Equation 2.50 can be used to estimate enthalpies of vaporization and sublima- 
tion if vapour pressure data are available, or to estimate vapour pressures from 
a value of the enthalpy change. Analysis of sublimation problems (see section 
7.4), is frequently difficult owing to the scarcity of published vapour pressure 
and enthalpy data. If two values of vapour pressure are available, however, 
a considerable amount of information can be derived from equation 2.50 as 
illustrated by the following example. 

Suppose that the only data available on solid anthracene are that its vapour 
pressure at 210° and 145 °C are 40 and 1.3mbar, respectively; the vapour 
pressure at 100 °C is required. Equation 2.50 can be used twice - first to 
calculate a value of AH S then that of the required pressure: 

,'40 \ AHJ483-4U 

1.3/ 8.314x483x418 

AH S = 88.5k.Imor 1 

Substituting this value of AH S in equation 2.50 again, 

/ 40 \_ 88.5 (483 - 373) 
11 V/>ioo°c/ ~ 8.314 x 483 x 373 


P\oo°c = 0.04 mbar 

2.12.1 Enthalpy of vaporization 

There are several methods available for the estimation of enthalpies of vapor- 
ization at the atmospheric boiling point of the liquid. Trouton's rule, for 
example, is only suitable for non-polar liquids, but the Giacalone equation is 
fairly reliable for both polar and non-polar liquids: 

AH wh = 88 T h (Trouton) (2.51) 

A// Vb = ( ^^ Vn^c (Giacalone) (2.52) 

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62 Crystallization 

where A// V b = enthalpy of vaporization (J moP ) at the boiling point, 7b = 
boiling point (K) of the liquid at 760 mm Hg, T c = critical temperature (K) 
of the liquid, and P c = critical pressure (atm) of the liquid. (1 atm = 
760mmHg= 1.013 x 10 5 N/m 2 .) 

The enthalpy of vaporization of a liquid at some temperature T\ can be 
calculated from its value at another temperature Ti by means of the Watson 

/ T T \0-38 

A# v i = A// v2 L° ~ A (2.53) 

For example, the enthalpy of vaporization of benzene at its boiling point 
(353 K) is 30.8 J mol, its critical temperature 563 K. From equation 2.44 the 
corresponding value at 25 °C (298 K) can be calculated as 33.6 J moP , which 
compares with an experimental result of 33.7 J moP . 

A critical account of these and other more recent methods is given by Reid, 
Prauznitz and Poling (1987). 

2.13 Heats of solution and crystallization 

When a solute dissolves in a solvent without reaction, heat is usually absorbed 
from the surrounding medium (still commonly referred to as the heat of 
solution), i.e. if the dissolution occurs adiabatically the solution temperature 
falls. When a solute crystallizes out of its solution, heat is usually liberated (still 
commonly referred to as the heat of crystallization) and the solution temper- 
ature rises. The reverse cases, viz. heat evolution on dissolution and heat 
absorption on crystallization, may be encountered with solutes that exhibit an 
inverted solubility characteristic, e.g. anhydrous sodium sulphate in water. 

The dissolution of an anhydrous salt in water at a temperature at which the 
hydrated salt is the stable crystalline form frequently leads to the release of heat 
energy, owing to the exothermic nature of the hydration process: 

AB+nH 2 0^ ABnH 2 

Table 2.9 lists the heats of solution of anhydrous and hydrated magnesium 
sulphate and sodium carbonate in water to illustrate the effect of water of 

The enthalpy changes associated with dissolution (A// so i) and crystallization 
(A// crys ) are generally recorded as the number of heat units liberated by the 
system when the process takes place isofhermally. According to this system 
of nomenclature, if an adiabatic operation is considered, the expression 
A// so i = +q (heat units per unit mass of solute) means that the solution 
temperature will increase; A// so i = — q means that it will fall. 

The magnitude of the heat effect accompanying the dissolution of solute in 
a given solvent or undersaturated solution depends on the quantities of solute 
and solvent involved, the initial and final concentrations and the temperature at 

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Physical and thermal properties 


Table 2.9. Heats of solution of anhydrous and hydrated salts in water at 
18 °C and infinite dilution 



Heat of solution, 
kJmor 1 

Magnesium sulphate 

Sodium carbonate 

MgS0 4 


MgS0 4 • H 2 


MgS0 4 • 2H 2 


MgS0 4 • 4H 2 


MgS0 4 • 6H 2 


MgS0 4 • 7H 2 


Na 2 C0 3 


Na 2 C0 3 • H 2 


Na 2 C0 3 • 7H 2 


Na 2 C0 3 • 10H 2 O 


which the dissolution occurs. The standard reference temperature is nowadays 
generally taken as 25 °C. 

The first differential heat of solution (heat of solution at infinite dilution), 
A/f^j, may be regarded as the heat liberated or absorbed when 1 mole of solute 
dissolves in a large amount of pure solvent. This is the value most generally 
recorded in the data handbooks. For inorganic salts in water, it normally lies 
between about 5 and 15kcalmoF , i.e. about 20 to 60kJmol~ 1 . For organic 
substances in organic solvents, it normally lies between 1 and 5kcalmoF' 
(about 5 to 20kJmol~ 1 ). The last differential heat of solution AHf ol is the 
amount of heat liberated or absorbed, when 1 mole of the solute dissolves in a 
large amount of virtually saturated solution. This is numerically equal to the 
heat of crystallization, A// crys , but of opposite sign. The relationship between 
these quantities is 



A/C, + AH dll = AH. 



In crystallization practice, however, it is usual to take the heat of crystal- 
lization as being equal in magnitude, but opposite in sign, to the heat of 
solution at infinite dilution, since this is the quantity most commonly available 
in the handbooks, i.e. 






Few values of heats of dilution are available in the literature, especially for 
the higher concentration ranges usually associated with industrial practice, but 
this quantity is usually only a small fraction of the heat of solution. Further- 
more, as the dilution of most aqueous salt solutions is exothermic, i.e. the 
concentration is endothermic, the true value of the heat of crystallization will 
be slightly less than that obtained by taking the negative value of the heat of 
solution alone. Therefore the calculated quantity of heat to be removed from a 
crystallizing solution will be slightly greater than the true value, and this small 
error can serve as a factor of safety in the design of cooling heat transfer 

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64 Crystallization 

An alternative way to evaluate the heat of solution, and hence to estimate the 
heat of crystallization, is to consider the effect of temperature on the solubility: 
the greater the effect the higher is the heat of solution of the solute as quantified 
by the relationship 

^~^> (256) 

where A.Hf ol is the final differential heat of solution. 

2.14 Size classification of crystals 

The most widely employed physical test applied to a crystalline product is the 
one by means of which an estimate may be made of the particle size distribu- 
tion. Product specifications invariably incorporate a clause that defines, often 
quite stringently, the degree of fineness or coarseness of the material. For many 
industrial purposes the demand is for a narrow range of particle size; regularity 
results in the crystals having good storage and transportation properties, a free- 
flowing nature and, above all, a pleasant appearance. Terms such as 'fine' and 
'coarse' are frequently used, although usually without definition, to describe 
crystalline and powdered materials. For pharmaceutical products, however, 
some guidance is available from recommendations in the British Pharmaco- 
poeia (2000), based on sieve gradings. 

Coarse all passes 1700 um >40% passes 355 um 

Moderately coarse all passes 710 um >40% passes 250 um 

Moderately fine all passes 355 um >40% passes 180 um 

Fine all passes 180 um >40% passes 125 um 

Very fine all passes 125 um >40% passes 45 um 

Microfine <90% passes 45 um 

Some of the more important procedures associated with the characterization 
of particulate solids are outlined below. 

2.14.1 Sampling 

The physical and chemical characteristics of a bulk quantity of crystalline 
material are determined by means of tests on small samples. These test samples 
must be truly representative of the bulk quantity; otherwise any results 
obtained will be grossly misleading or completely useless. Inefficient sampling 
followed by careful analysis in the laboratory constitutes a waste of everyone's 
time and effort. Sampling, which is a highly specialized skill, should be carried 
out by conscientious, well-trained personnel who are fully aware of the tests 
that are to be made on the sample, without having any direct interest in the 
outcome of the analyses. 

The actual technique employed for sampling will depend on the nature of the 
bulk quantity of material, its location, the properties to be tested, the accuracy 

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Physical and thermal properties 65 

required in the test, and so on. Difficulties may be encountered in the sampling 
of solids in containers after transportation, owing to the partial segregation of 
fine and coarse particles; the fines tend to migrate towards the bottom of the 
container, and thorough remixing may be the only answer. Similar problems 
caused by segregation may be met in the sampling of solids flowing down 
chutes or through outlets. 

Hand sampling, widely employed for batch-produced or stored materials, is 
time-consuming and prone to error, but its use cannot always be avoided. One 
common method involves the use of a sample 'thief, a piece of pipe with 
a sharp bottom edge which is plunged into the full depth of the material. It is 
then withdrawn and the sample removed. This operation can be performed at 
fixed or random intervals in the bulk quantity. Sampling at intervals by means 
of scoops or shovels, known as grab sampling, is also widely used, but serious 
errors can be encountered when dealing with non-homogeneous materials. 

Automatic sampling is preferred to sampling by hand, and is also better 
suited to continuous processes. Ideally, the sample should be taken from 
a moving stream of solids or slurry. To eliminate segregation effects, samples 
should be taken by collecting the whole of the flowing stream for short periods. 
Isokinetic sampling is recommended for crystal suspensions in agitated vessels 
and pipelines (section 9.2). 

Bulk samples, which may range up to several hundred kilograms for large 
tonnage lots, have to be reduced to a smaller laboratory sample which, in turn, 
will have to be divided into several smaller test samples for subsequent analysis. 
These operations may be carried out by hand or with the aid of a sample divider. 

The best-known hand method is that of coning and quartering. The sequence 
of operations, carried out on a clean, smooth surface, or on glossy paper for 
small quantities in the laboratory, is shown in Figure 2.11. The bulk sample is 
thoroughly mixed and piled into a conical heap. The pile is then flattened and 
the truncated cone divided into four equal quarters (Figure 2.11c). This may be 
done, for example, with a sharp-edged wooden or sheet metal cross pressed into 
the heap. One pair of opposite quarters are rejected, the other pair are thor- 
oughly mixed together and piled into a conical heap, the procedure being 
repeated until the required laboratory sample is obtained. 

A simple sample divider is the riffle which usually takes the form of a box 
divided into a number of compartments with bottoms sloping about 60° to the 
horizontal, the slopes of alternate chutes being directed towards opposite sides 
of the box. Thus, when the bulk sample is poured through the riffle, it is divided 


(a) (b) (c) (d) 

Figure 2.11. Method of coning and quartering: (a) hulk sample in a conical heap; (h) 
flattened heap; (c) flattened heap quartered; (d) two opposite quarters mixed together and 
piled into a conical heap 

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66 Crystallization 











— v 

Figure 2.12. Battery of riffles used for reduction of a large quantity of material 

into two equal portions. The dimensions of a riffle depend on the size of the 
particles and the quantity of material - an even flow of solids must be spread 
over the whole inlet area. When very large bulk samples have to be reduced to 
small test quantities, a battery of riffles decreasing in size may be employed, as 
shown diagrammatically in Figure 2.12. This arrangement is also suitable for 
the continuous or intermittent sampling of materials flowing out of hoppers or 
other items of process plant. 

The rotary sample divider, or spinning riffle, is less prone to operator error than 
is the static riffle. Basically it consists of a hopper which allows particulate material 
to flow on to a vibrating chute which then discharges into a number of sample boxes 
located in a rotating ring. Several units of this type are available commercially. 

The statistical theories of sampling are discussed by Allen (1990) who also 
describes, in considerable detail, a large number of sampling methods. BS 3406/1 
(1986) also gives guidance on sampling, the sub-division of laboratory samples 
and the reporting of results. 

2.14.2 Particle size and surface area 

A large number of methods are now available for measuring particle size, some 
of which are listed in Table 2.10 together with their approximate ranges of 
application. The book by Allen (1990) and a series of British Standards (BS 
3406, 1986; BS 4359, 1984) give excellent coverage of the subject. Only a short 
review will be given here. 

It is not possible to measure or define absolutely the size of an irregular particle, 
and perfectly regular crystalline solids are rarely, if ever, encountered. The terms 
length, breadth, thickness or diameter applied to irregular particles are mean- 
ingless unless accompanied by further definition, because so many different values 
of these quantities can be measured. The only meaningful properties that can be 
defined for a single solid particle are the volume and surface area, but even the 

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Physical and thermal properties 67 

Table 2.10. Some methods of particle size measurement and their aproximate useful size 

Method Size range 


Sieving (woven wire) 125 000-20 
Sieving (electroformed) 120-5 

Sieving (perforated plate) 125 000-1000 
Microscopy (optical) 150-0.5 

Microscopy (electron) 5-0.001 

Sedimentation (gravity) 50-1 

Sedimentation (centrifugal) 5-0.1 

Electrical sensing zone (Coulter) 200-1 

Laser light scattering (Fraunhofer) 1000-0.1 

Permeametry 1 Surface area measurement: useful for particle sizes smaller than 
Gas adsorption/ about 50 urn 

measurement of these quantities may present insuperable experimental dif- 
ficulties. All particle size measurements are made by indirect methods: some 
property of the solid body which can be related to size is measured. 

Despite these difficulties of definition and measurement it is most convenient 
for classification purposes, if a single-length parameter can be ascribed to an 
irregular solid particle. The most frequent expression used in connection with 
particle size is the 'equivalent diameter', i.e. the diameter of a sphere that 
behaves exactly like the given particle when submitted to the same experimental 
procedure. Several of these equivalent diameters are defined below. 


Woven wire test sieves were formerly designated by a mesh number (the 
number of wires per inch) but as the important sieve characteristic is the size 
of its apertures all standard test sieves are now designed, by international 
agreement, by their aperture size in millimetres or micrometres. The aperture 
sizes in a standard series are related to one another, e.g. following a fourth root 
of two (1.189) or a tenth root often (1.259) progression. The two most widely 
used standard sieve scales are the American (ASTM Ell, 1995) and British (BS 
410, 2000) both of which are compatible with the international scale (ISO 3310, 
2000) (Table 2.11). 

The range of aperture sizes in most standard series extends from 125 mm to 
20 um. At the top end of the range particles must be carefully hand-placed on 
the sieve. At the lower end, sieving with the aid of a liquid is often needed to 
assist the flow of particles through the mesh. Particles that pass through a sieve 
are characterized by an equivalent sieve aperture diameter, d s a ., the diameter of 
a sphere that would just pass through. Care needs to be taken to interpret this 
quantity, however, as explained in section 2.14.3 (see Figure 2.14). 

Perforated plate sieves are available, with round (125 to 1 mm) or square (125 
to 4 mm) apertures, for coarse particle sizing. Microsieves with electroformed 

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68 Crystallization 

Table 2.11. Comparison between the British and US standard wire mesh sieve scales* 
showing the danger of using the 'mesh number' as a sieve designation 

Aperture width (|-im) 






















3 1 

J 2 




































































































*Both standard test sieve scales BS 410 and ASTM Ell are compatible with the international (ISO 
3310) scale. 

f The definition of 'mesh number' is the number of apertures per inch in the sieve mesh. This 
obsolete designation leads to confusion because different standards specify different wire diameters. 

round or square apertures (120 to 5 urn) in nickel plate are available for very 
fine particle sizing. 

Sieving is basically a very simple and justifiably popular particle sizing 
technique, but the precautions necessary to produce reliable data do not appear 
to be widely appreciated. Some of the more important points to note about the 
use of standard test sieves for particle size analysis are as follows. 

1. Particles must not be forced through the sieve apertures. 

2. Sieving should be continued to an end-point, i.e. until the amount of 
material passing through ceases to affect the result significantly. When using 
a mechanical shaker, it is recommended that each sieve removed from the 
stack should be given a brief brisk tapping and shaking by hand to ensure 
that the end-point has been reached. If it has not, sieving must be continued. 

3. It should be clearly understood that the aperture size marked on the sieve is 
only a nominal size. The actual value can vary from this value, within 

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Physical and thermal properties 69 

specified tolerances. For accurate work, it is advisable to calibrate the sieve, 
e.g. by sieving a reference sample of known size distribution. 

Procedures for test sieving, both wet and dry, are prescribed in standard 
specifications (ISO 2591, 1989; BS 1796, 1990). Methods for the analysis of 
sieve test data are described in section 2.14.4. 


Microscopy is commonly used as a basic reference method for particle sizing 
since individual particles may be observed while measuring or assessing their 
size, shape and composition. Particle images may be viewed directly in an 
optical microscope or by projection. The particle size may be recorded as the 
projected area diameter <i p . a ., the diameter of a circle that has the same area as 
the projected image of the particle viewed in a direction perpendicular to its 
plane of maximum stability. This may be assessed by comparison with grad- 
uated circles on an eyepiece graticule. The microscopic method can be tedious 
and time-consuming, although automatic counting devices are now available. 
Photographic methods are popular, but can introduce further errors into the 
system (BS 3406/4, 1990; ISO 13322, 2001). The problems of preparing a micro- 
scope slide containing a well-dispersed representative sample of small crystals 
can be very considerable (Allen, 1990). 


A simple sedimentation technique, which readily lends itself to the determina- 
tion of crystal size distribution in the range 1-50 um, is the Andreasen pipette 
method. Although it is generally better to prepare a fresh suspension of the 
crystals under test in a suitable inert liquid, it is possible to classify crystals 
suspended in their own mother liquor. If the difference in density between the 
particles and suspending liquid is <0.5gcm~ 3 special care must be taken to 
avoid convection currents. The method, briefly, is as follows (BS 3406/2, 1986). 

A homogeneous suspension of the crystalline material in a suitable liquid is 
prepared in the graduated sedimentation cylinder of capacity ^600 cm 3 (Figure 
2.13). Small samples (e.g. 10 cm 3 ) of the suspension are withdrawn through the 
fixed pipette, at a known depth, h, below the liquid level, at chosen time 
intervals. The samples, including the one taken at zero time, are analysed for 
total suspended solids content by a suitable method. Ideally the suspension 
should be dilute (< 3 per cent) and a dispersion agent may be needed to prevent 
agglomeration: for particles in insoluble water a 0.1 per cent solution of sodium 
pyrophosphate is generally suitable. 

A sample taken at time t will contain no particles larger than size <fst 
calculated from Stokes' law which may be written 

[ Wjr, 1 1/2 
XPs ~ Pf)gt_ 

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70 Crystallization 

10 ml 

rdH 20 cm 

-;— 10 cm 

Figure 2.13. Andreasen fixed pipette method 

Thus by taking samples at suitable intervals, e.g. 0, 5, 10, 20, 40, 80 . . . min, the 
size distribution of the original suspension may be evaluated. For routine 
analysis only one or two samples may be needed to characterize the particles. 
If t is measured in minutes, h in cm, p in gcm~ 3 and n in centipoise, then the 
particle size d, in urn, is given by 

cfet = 17.5 


.(Ps - Pf)t. 



For a given sample, n, the cumulative mass percentage, P n , of particles 
smaller than the limiting Stokes' diameter for the time interval, t n , may be 
calculated from the mass W„, of the suspended solids in the fraction by 

P„ = 100 

Wn V_ 

w ' v„ 


where W = mass (g) of solids originally suspended in the apparatus, V = 
original volume of the suspension (cm 3 ) and V„ = volume of sample taken 
via pipette (cm 3 ). 

A typical analysis is given in Table 2.12, where the size distribution of 
precipitated calcium carbonate (p s = 2.7 gem -3 ) is measured by sedimentation 
at 20 °C in water containing 0.1 per cent sodium pyrophosphate as dispersant 
(p f = l.Ogcirr 3 and r\ = 1.0 cP, i.e. 10~ 3 Ns/m 2 ). In this test the CaC0 3 was 
determined volumetrically by adding 0.2 M HC1 to each sample, boiling to 
remove CO2 and back-titrating with 0.1 M NaOH. A 'blank' was run on the 
suspending liquid. Alternatively, in this case, a gravimetric method could have 
been used, i.e. by evaporation to dryness. 

Descriptions of other gravitational sedimentation techniques are outlined in 
a recent international standard (BS ISO 13317, 2000) for particles in the size 

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Physical and thermal properties 71 

Table 2.12. Measurement of the particle size distribution of a sample of precipitated 
calcium carbonate by the Andreasen pipette method 




CaC0 3 in 








h (cm) 

d (nm) 

W„ (g) 









































*Sample volume V„ = 10cm 3 . 

fTest sample mass W = 14.3 g in suspension volume V = 620cm 3 . 

range 1-100 um. For particles smaller than about 5 |im, however, problems can 
arise from convection effects and Brownian motion, but these difficulties may 
be reduced by speeding up the settling process by centrifuging the suspension. 
A number of procedural methods and commercial equipment for centrifugal 
sedimentation are now available for determining particle size in the 0.1 — 5 (xm 
range (Allen, 1990; ISO 13318, 2000). 

Electrical sensing zone (Coulter) methods 

In the Coulter technique, particles have to be suspended in an electrolyte 
solution and then induced to pass through a small orifice, with surrounding 
electrodes, located in the measurement cell. Changes in electrical impedance in 
the orifice channel for each particle passage are measured and counted. The 
result is a number-size (volume) distribution of particles (BS ISO 13319, 2000). 
The method has found applicability in a wide range of industries. For applica- 
tion to crystallizing systems, however, it is important to choose a unit in which 
the voltage between the inner and the outer electrode is automatically adjusted 
so as to maintain a constant current. This renders the calibration, and hence the 
actual counting, insensitive to the type of electrolyte used as well as to concen- 
tration and temperature changes within the electrolyte. This precaution is of 
paramount importance for crystallization studies where changes in electrolyte 
properties due to phase transitions are inevitable (Jancic and Grootscholten, 1984). 

Laser light scattering (Fraunhofer) methods 

Another widely used particle size analyser is based on the forward scattering of 
laser light through a dilute (< 1 % by volume) suspension of crystals retained in 
a small (~10mL) agitated cell. The resulting Fraunhofer diffraction pattern is 
detected and translated, by means of the instrument software, into a particle 
size distribution (BS ISO 13320, 2000). 

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72 Crystallization 

Specific surface area measurement 

In many cases, particularly for very small particles, surface area is a more 
appropriate characteristic to assess than some size based on an equivalent 
diameter. Particle surface area is important, for example, in paints and pig- 
ments or when chemical reactivity is an important property, as in the setting of 
cement. Precipitated materials are often characterized in this manner. Amongst 
the several techniques available, those based on permeability and gas adsorp- 
tion are probably the most popular. 

In the permeability methods a known quantity of air is forced through a small 
bed of the fine solids under a constant pressure drop, and the flow time is 
recorded. The theory is based on the laminar flow of fluids through porous 
beds, and the specific surface area S (m 2 g _1 ) of the material is calculated from 
the Kozeny equation 

s 2 = v^r-2-^-2 ( 2 - 6 °) 

kurjLp 2 (l-e) 2 

where AP = pressure drop across the bed; e = voidage of the bed; L = depth of 
the bed; r\ = viscosity of the air; u = empty-tube velocity; p = density of the 
solid material; k is a constant (Kozeny's constant), which has a value equal to 
about 5.0 for granular solids. Several different types of permeability cell are 
available (BS 4359/2, 1982; Allen, 1990). 

A solid particle exposed to a gas will adsorb gas molecules on to its exposed 
surfaces. The derivation of a multilayer adsorption theory for gases on solid 
surfaces by Brunauer, Emmett and Teller in 1938 led to the development of the 
so-called BET adsorption methods for measuring the specific surface area of 
particulate solids. Several techniques are available (BS 4359/1, 1982; Lowell 
and Shields, 1984; Allen, 1990). 

Dry tests like the BET adsorption method can often give misleading informa- 
tion when used to characterize precipitated materials because the sample 
preparation operations of filtering, washing and drying can result in consider- 
able damage and distortion to the particles. For this reason, a dye adsorption 
technique has been found to give a more realistic measurement of the specific 
area of precipitates while still suspended in their original mother liquor (Mullin 
et cil., 1989a). A sample of the precipitate suspension is pipetted into a small 
flask containing a known quantity of a concentrated solution of a suitable dye, 
the selection of which depends on the chemical nature of the precipitate. After 
shaking to allow the mixture to come to equilibrium it is clarified in a labor- 
atory centrifuge and the clear liquid analysed with a UV spectrophotometer. 
Knowing the mass of dye adsorbed on a given mass of precipitate, and the 
'coverage' value of the dye (the so-called Paneth value), it is a straightforward 
matter to calculate a specific surface area of the precipitate in, for example, 

2 -1 

m g . 

It should also be noted that size data produced by the many different 
electronic techniques and instruments now available are dependent on the 
particular analysis algorithm incorporated into the instrument by the manu- 
facturer. Significant differences in sizing results can therefore be recorded by 

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Physical and thermal properties 73 

commercial equipment of basically the same type. Indeed, for precise work, 
repeatability tests should be carried out to determine the closeness of agreement 
between independent tests made with the same method on identical test mater- 
ial in the same laboratory by the same operator using the same equipment. For 
inter-laboratory investigations reproducibility tests should be made to determine 
the closeness of agreement between tests made with the same method on 
identical test material in different laboratories with different operators using 
different equipment. Recommended procedures for repeatability and reprodu- 
cibility tests are described in ISO 5725 (1992). 

Comparison of data 

From the above brief descriptions of a few commonly used particle sizing 
techniques it can be seen that the different methods arrive at an expression of 
particle 'size' after measuring quite different particle properties. It is not 
surprising, therefore, that for a given sample of particulate material the differ- 
ent techniques will give different, often very different, sizes. 

It is advisable to use, if at all possible, only one sizing technique over the 
whole particle size range encountered in any one analysis. Even in sieving, only 
one type of sieving medium should be used throughout (woven wire, perforated 
or electroformed plate, round or square aperture) because of their individual 
sieving characteristics. When it is necessary, because of a very wide size dis- 
tribution, to apply two different sizing techniques it is advisable that the data 
from both methods overlap over a significant part of the range to enable all the 
data to be converted to a common basis. Examples of overlap studies have been 
reported for sieve/zone sensing (Mullin and Ang, 1974; Jancic and Grootschol- 
ten, 1984) and for sieve/laser light scattering (Brecevic and Garside, 1981). 

2.14.3 Shape factors 

A precise calculation of the volume or surface area of a solid body of regular 
geometric shape can only be made when its length, breadth and thickness are 
known. For particulate solids in general, these three dimensions can never be 
precisely measured. Therefore, before a brief account is given of some of the 
methods of calculation available, a word of warning is necessary. It must be 
fully appreciated that the precision of calculation is always far greater than that 
of measurement of the various quantities used in the mathematical expressions. 
An equation, especially a complex one, always has a look of absolute depend- 
ability, but in this particular connection it most certainly leads to a false sense of 
security. All calculated volume or surface area data must be used with caution. 
Most calculation methods are based on one dimension of the particle, usually 
the equivalent diameter. If this dimension is obtained from a sieve analysis, it will 
be the sieve aperture diameter, d sa ,; but as crystals are never true spheres, this 
diameter will normally be the second largest dimension of the particle. Figure 2.14 
demonstrates some particle shapes that would, in a sieve analysis, all yield the 
same value for d s a . . One potential source of error is thus clearly seen. 

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Screen* jt % |9 


^ P 


Figure 2.14. Various particle shapes that would all be classified under the same 
sieve-aperture diameter 

For a single particle, the size of which is defined by some length parameter 
or equivalent diameter, d, and density, p the following relationships can be 


surface area 

--f v d 3 

'-fvpd 3 

-fsd 2 



The constants /,, and f s may be called volume and surface shape factors, 
respectively. In the expressions for crystal growth rate in section 6.2.1, fy and 
fs are given the symbols a and f3, respectively. These latter symbols are also 
used in Chapter 8. 

For spherical (diameter = d) and cubic (length of side = d) particles 


a = f v = — (sphere) and 1 (cube) 

j3 = f s = -k (sphere) and 6 (cube) 

The shape factors are readily calculated for other regular geometrical solids. 
For an octahedron, for example, with d representing the length of an edge, 
v = V(2)rf 3 /3 and s = 2y/(3)d 2 , therefore 



■-v/d 3 

S/d 2 = 

: V(2)/3 = 0.471 

2^3 = 3.46 

From equations 2.61-2.63 two basic ratios may be defined: 

_fd^ = F 

v f v d 3 d 

s _ f s d 2 F 
m fpd 3 pd 

surface : volume 

surface : mass 


Equation 2.65 defines the useful quantity known as the specific surface area, 
i.e. the surface area per unit mass of solid. The constant F(=f s /f v = (3/a) may 
be called the overall, surface-volume or specific surface shape factor. For 
spheres and cubes, F = 6. For other shapes F > 6. For an octahedron 
f = \/(2)/3, f s = 2\/3 and F = 7.35. Values of F ~ 10 are frequently encoun- 
tered in comminuted solids, and much higher values may be found for flakes 
and plate-like crystals. If the particles are elongated or needle-shaped, their 
volume and surface area may be calculated on the assumption that they are 

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Physical and thermal properties 75 

cylindrical; length and diameter may be measured microscopically, or the 
diameter can be taken as the equivalent sieve aperture diameter. 

For example, a crystal with a length : breadth : thickness ratio of 5:2:1 
would be characterized in sieving by its second largest dimension, i.e. d = 2. 
Therefore/,, = v/d 3 = 10/8 = 1.25, f s = s/d 2 = 34/4 = 8.5 and F =f s /f r = 6.8. 

Volume shape factors may be measured by weighing a known number of 
particles from a close-sieve fraction, the number depending on the size of the 
particles and the accuracy with which the total mass can be weighed. The need 
for a close-sieve fraction arises from the fact that the shape factor of crystals 
can vary greatly with size (Garside, Mullin and Das, 1973). 

The determination of volume shape factors for particles smaller than about 
500 (im becomes extremely difficult, since it may be necessary to count and 
weigh several thousand particles. However, the following method may be used 
to simplify the procedure. Prepare a sample of the particles by sieving between 
two close sieves. Clean the finer of the two sieves (the retaining sieve) and 
attach a strip of adhesive tape of known mass and dimensions, to its underside. 
Place a quantity of the particles on the sieve and shake the sieve for several 
minutes. Peel off the adhesive tape, which will now have hundreds or thousands 
of particles in a regular matrix (more or less one per sieve aperture). The 
approximate number of particles per unit area can be determined from the 
designation of the sieve mesh. For example, a 150(im aperture sieve with 
100 um diameter wires contains about 1600 apertures per cm 2 . The adhesive 
strip can then be weighed and the average mass of one particle determined. 

Surface shape factors are much more difficult to measure than volume shape 
factors and they are subject to greater uncertainty. One method is as follows. 
A few individual crystals are observed through a low-power microscope fitted 
with a calibrated eyepiece, and sufficient measurements taken to allow a sketch 
to be made of a representative geometric shape, e.g. a parallelepipedon, ellips- 
oid, oblate spheroid, etc. The surface area of the representative solid body 
may then be calculated. It should be appreciated, of course, that the result of 
such a calculation will be prone to significant error. 

Another quantity that has been used to characterize crystal shape is 
the sphericity, ip, defined as the ratio of the surface area of a sphere having 
the same volume as the particle to the apparent estimated surface area of the 
particle. This can be rewritten (Nyvlt, 1990) as 


For isometric particles ip is close to 1 while for needles or platelets its value is 
much lower. Evaluation of ip is useful for checking the values of/V and./s since 

< V< 1- 

When three mutually perpendicular dimensions of a particle may be deter- 
mined, Heywood's ratios may help to characterize shape (Allen, 1990): 

elongation ratio n = LjB 
flakiness ratio m = BjT 

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76 Crystallization 

The thickness T is the minimum distance between two parallel planes which 
are tangential to opposite surfaces of the particle, one plane being the plane 
of maximum stability. The breadth B is the minimum distance between two 
parallel planes which are perpendicular to the planes defining the thickness 
and are tangential to opposite sides of the particle. The length L is the distance 
between two parallel planes which are perpendicular to the planes defining 
thickness and breadth and are tangential to opposite sides of the particle. 

When using shape factors, it is important to remember that they depend on 
the dimension chosen to characterize the particle. For example, a geometrical 
shape such as regular parallelepipedon with a length : breadth : thickness ratio, 
L : B : T, of 5 : 2 : 1 the following shape factors may be calculated: 

/v /. F 

using L= 5 0.08 1.4 17 

B=2 1.25 8.5 6.8 

T=\ 10 34 3.4 

Different answers again would result if measured values of the different equi- 
valent diameters, such as <i sa ., c/ p . a ., <ist, etc. were used. An example of apparent 
shape factor change caused by the use of different sieves (woven wire, round 
or square hole, perforated plate) to measure the particle size is reported by 
Garside, Mullin and Das (1973). 

2.14.4 Size data analysis 

Mean particle size 

In a total mass, M, of uniform particles, each of mass m and equivalent 
diameter d, the number of particles, n, is given by 

M M n <<n 

n = — = — — j (2.67) 

m f v pd i 
and the total surface area, Y,s, by 

f s Md 2 FM 

Y,s = ns= — — r = -— (2.68) 

Jvpd* pd 

Equations 2.67 and 2.68 can be applied to masses of non-uniform particles if 
a suitable average or mean value of d can be chosen. This becomes an intractable 
problem, however, since there are so many possible choices that could be made. 
The simplest of all average diameters is the arithmetic mean. For example, if 
sieving has been carried out between two sieves of aperture a\ and a.2, the 
average particle equivalent diameter is given by 

da = (fli + ai)\l (2.69) 

This description may be quite adequate for two close sieves in a \Jl series, but it 
can be absolutely meaningless for two sieves at extreme ends of the mesh range. 
Another simple average diameter is the geometric mean, defined by 

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Physical and thermal properties 11 

d g = y / (aia 2 ) (2.70) 

Values of d calculated from equation 2.70 are smaller than those given by 
equation 2.67, but for two close sieves the difference is not great. 

The volume mean diameter (or mass mean if the particle density is constant) 
is widely used: 

- End 4 E(Md) 

dv = ^P = ^M- (2 - 71) 

When the surface area of the particles is an important property the surface 
mean diameter can be employed, defined by 

- _ Ena* _ EM 

s ~ En~S " E(M/rf) ( } 

where n and M are the number and mass, respectively, of all particles of 
equivalent diameter d. 

The root mean square diameter is also frequently used when surface proper- 
ties are important. This statistical quantity is defined by 

7 / (End 2 \ 

V E« / V [E(M/d 3 ) 



Values of the overall mean diameters calculated from equations 2.71-2.73 
can differ considerably (see Table 2.13), yet for a mass of particles with a wide 
size distribution there is no general agreement as to the preferred method. 

Two other statistical diameters are often encountered, viz. the modal and 
median diameters; both are determined from frequency plots (size interval 
versus number of particles in each interval). The modal diameter is the diameter 
at the peak of the frequency curve, whereas the median diameter defines a mid- 
point in the distribution - half the total number of particles are smaller than the 
median, half are larger. If the distribution curve obeys the Gaussian or Normal 
Error law, the median and modal diameters coincide. 

Table 2.13. Calculation of overall 'mean' diameters 

Size range 

Mean size 

Mass of 



M/d 3 


of fraction, 







0.031 x 10-" 






0.139 x 10-" 




13 900 


0.812 x 10-" 






1.353 x 10-" 






1.417 x lO" 6 




3.752 x 10-" 

d, = 39 300/100 = 393 nm. 
d s = 100/0.2941 =340 urn. 
J rms = (0.2941/3.75 x 10" 6 ) I/2 = 280 urn. 

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78 Crystallization 

In connection with particle size measurement more than 20 different 'mean' 
diameters have been proposed at one time or another; and while several have 
certain points in their favour in special cases, none has yet been found to be 
generally satisfactory. Therefore all calculations based on an average diameter 
are prone to appreciable error, and it is recommended that such calculated 
quantities be clearly annotated with the method of calculation so that the 
results of different workers can be compared. 

Graphical analysis 

Once a size analysis has been performed and the results recorded, there remains 
the task of assessing the size characteristics of the tested material and of extract- 
ing the maximum amount of information from the data. While a table may record 
all the measured quantities, this form of expression is not always the best one; the 
magnitudes of the various quantities may be readily visualized, but certain trends 
may be completely obscured in a mass of figures. The real significance of a sizing 
test can most readily be judged when the data are expressed graphically. From 
such a pictorial representation trends in the data are easily detected, and the 
prediction of the expected behaviour of the material on sieves other than those 
used in the test can often be made with a reasonable degree of accuracy. 

Many different forms of graphical expression may be employed, and the use 
and applicability of some of these methods are demonstrated below with 
reference to the results of a sieve test. The graphical procedures described, 
however, are applicable, with suitable nomenclature changes, to all methods 
of particle size analysis. The sieve test data in Table 2.14 are recorded in three 
different ways, viz. the percentage by mass of the fractions retained on each 
sieve, and the cumulative mass percentages of oversize and undersize material. 

Four types of graph paper are commonly used for plotting particle size 
distributions, depending on the sort of information that is required: (a) ordinary 

Table 2.14. Sieve test data used for the construction of Figures 2.15 to 2.18 










per cent 

per cent 

per cent 
















































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Physical and thermal properties 


squared or arithmetic, (b) log-linear or semi-log, where one of the axes is 
marked off on a log scale and the other on an arithmetic scale, (c) log-log where 
both axes are marked off on a logarithmic scale, and (d) arithmetic-probability, 
where one axis is marked off on a probability scale, the intervals being based on 
the probability integral. Log-probability and double-logarithmic (RRS) grids 
also find use in special cases. 

In Figure 2.15a the mass percentages of the fractions retained on each 
successive sieve used in the test are plotted against the widths of the sieve 
apertures (in microns). The lines joining the points have no significance; they 
merely complete the frequency polygon. The sharp peak in the distribution 
curve occurs at 850 um. This point, however, represents the fraction that passes 
the 1180um sieve and is retained on the 850 um, so it could be plotted at the 
'mean' size of 1015 um. Alternatively, the results may be represented in the 
form of a frequency histogram {Figure 2.15b) depicting the size range of each 
collected fraction. From both diagrams the general picture of the overall spread 
of particle size can be seen quite clearly, but the simple arithmetic method of 
plotting suffers from the disadvantage of producing a congested picture in the 
regions of the fine mesh sieves. 

When the data are plotted on semi-log paper (Figure 2.16a), with the aperture 
widths recorded on the logarithmic scale, the points in the coarse sieve region are 
brought closer together, and those in the fine sieve region located further apart 
than in the corresponding simple arithmetic plot. In fact, the successive points 
are more or less equally spaced along the horizontal scale. The frequency 
histogram (Figure 2.16b) is composed of columns of approximately equal widths. 

Corresponding B.S mesh numbers Corresponding B.S. mesh numbers 



36 25 





1 II 

1 1 





£ 25 





•£ 20 






o- 15 




5 10 




o 5 



i 1 

500 1000 1500 2000 2500 

Sieve aperture width, ^m 



36 25 18 14 

O 7 


1 1 



c 20 





°- 15 



* 10 




Z 5 






500 IO00 1500 2000 2500 

Sieve aperture width, ^.m 

Figure 2.15. Sieve test data plotted on arithmetic graph paper - percentage by weight of 
the fractions retained between two given sieves in the B.S. series: (a) the frequency polygon; 
(b) the frequency histogram 


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Corresponding B.S. mesh numbers 

150 100 72 52 36 25 

Corresponding B.S. mesh numbers 

50 100 72 52 36 25 18 14 10 7 

l — r 


100 i0 ° i 00 5 °° 1000 200 ° soo ° 
Sieve aperture width, ^.m 


200 SOO 500 1000 200 ° soo ° 

Sieve aperture width, ^.m 

Figure 2.16. Sieve test data plotted on semi-log graph paper - percentage by weight of 
fractions retained between two sieves in the B.S. series: (a) the frequency polygon; (h) the 
frequency histogram 

The semi-log graphs of the cumulative oversize and undersize percentages 
(Figure 2.17a) show that the curves are mirror images of each other. In practice 
only one need be plotted. The two curves cross over at the median size (50 per cent 
is larger than the median, and 50 per cent is smaller). In this case the median 
size is 870 (im. Interpolation is facilitated by the even spread of the plotted 
points. It can be estimated, for instance, that about 87 per cent of the original 
material would be retained on a 355 nm sieve or that about 7 per cent would 
pass through a 250 |J.m sieve. 

The cumulative percentages of oversize and undersize particles may also be 
plotted against aperture size on a log-log basis (Figure 2.17b). In this type of 
plot the cumulative undersize data tend to lie on a straight line over a wide 
range of particle size, about 100 to 1200 nm in this case. The undersize and 
oversize curves are clearly not mirror images, and oversize data are rarely 
correlated on this basis. 

The log-log method of plotting of undersize data is extremely useful because 
rough checks may be made on the size distribution by the use of only two, or 
possibly three, test sieves. Material of the type considered in Table 2.11 could be 
size-checked with 1000, 500 and 250 urn sieves, for example. 

Coefficient of variation (CV) 

Probability plots have often been suggested for particle size analysis, par- 
ticularly in connection with the assessment of comminution processes. 

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Physical and thermal properties 81 

Corresponding B.S.mesh numbers 

150 100 72 52 36 25 18 14 10 


200 300 400 500 1000 

Sieve aperture width, pm 

2000 3000 


Corresponding B.S.mesh numbers 

150 100 72 52 36 25 18 14 10 7 

100 200 300 400 1000 2000 3000 

Sieve aperture, ^.m 


Figure 2.17. Sieve test data plotted on (a) semi-log graph paper, (b) on log-log graph 
paper-cumulative oversize and undersize percentages 

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82 Crystallization 

The following method, proposed by Powers (1948) for use in the sugar industry, 
has much to be said in its favour for the size specification of crystalline products. 
This method employs arithmetic-probability graph paper (one scale divided 
into equal intervals, the other marked off according to the probability integral) 
and provides a simple means of recording the crystal size distribution in terms 
of two numbers only - the median size (MS), and a statistical quantity the 
coefficient of variation (CV) expressed as a percentage. 

The evaluation of MS/CV is demonstrated in Figure 2.18a with the sieve 
analysis data from Table 2.13. The cumulative undersizes (or oversizes if 
preferred) are plotted on the probability scale, the sieve aperture sizes on the 
arithmetic scale. If the data between about 10 and 90 per cent lie on a straight 
line, the MS/CV method can be applied. The data in Figure 2.18a comply with 
this requirement. Thus the median size is 870 (im. The coefficient of variation 
can be deduced as follows. 

The equation for the normal probability (Gaussian) curve may be expressed 


(d - df 

2a 2 


where d is an equivalent particle diameter, in this case based on sieve aperture 
size, and a is the standard deviation. If the area enclosed under the normal 
curve between sieve apertures d = to oo is taken as unity, the area enclosed 
between d = and d = d + a, where d is the median (50 per cent) size MS, is 
0.8413. This value is obtained from tables of the normal probability function. 
Therefore the area enclosed between d = d + a and d = oo is 1—0.8413 = 
0.1587. The value of a, the standard deviation, can be obtained from the 
arithmetic probability diagram by reading the value of d at 84.13 per cent 
(84 per cent is accurate enough for this purpose) and subtracting the value of 
d. Alternatively, the value of d at 15.87 (or 16) per cent can be subtracted from 
d, i.e. 

u = ^84»/ — d = d — d\(,% 

These two values of a may not coincide, so a mean value can be taken as 

^84% — d\(y«/„ 
° = 2 

The coefficient of variation, as a percentage, is given by 

cv= ioo. = ioo. 

d «50% 


CV = 100(J84% ~ dl6%) (2.76) 


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Physical and thermal properties 83 



« 98 



T> _ 

c 90 


1 *> 

I 70 

< i 

& 60 
«. 50 

I50%_ _ _/ 




| 30 

/ 1 


« 20 

^ l6% _/ 

o 10 

- 7" ' 


' i i 

F 5 

- ° *- 1 1 


° 2 


icj? 7 ^! 12 ^! , 


500 1000 1500 2000 

Sieve aperture^im 


100 200 500 1000 3000 

Sieve aperture, ^m 

Figure 2.18. Sieve test data plotted on (a) arithmetic! probability graph paper {cumulative 
under size percentages) illustrating the use of the MS/CV method of analysis, (b) on Rosin- 
Rammler-Sperting (RRS) double logarithmic graph paper 

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84 Crystallization 

From Figure 2.18a 

100(1270 - 440) 

CV = = 48 per cent 


and the size distribution can be specified, in terms of MS/CV, as 870/48. 

The size distribution does not have to be Gaussian for the MS/CV method to 
be applied. Many skew distributions also give the necessary linear relationship 
between about 10 and 90 per cent, although in these cases MS will not coincide 
with the modal diameter at the peak of the distribution curve. 

For skew distributions that are not approximately Gaussian over the 10-90 
per cent region, plotting on log-probability paper (one scale logarithmic, the 
other marked off according to the logarithmic probability function) may give a 
better correlation. The equation of the logarithmic normal curve is 

f{\ogd) = — — exp 

loger v2vt 

(log d -log d'f 

2 log 2 a' 


where d' is the geometric mean size and a 1 is the geometric standard deviation. 

The log-normal distribution gives a curve skewed towards the larger sizes, 
and it frequently gives a good representation of particle size distributions from 
precipitation and comminution processes. Furthermore, the log-normal distri- 
bution is often used because it overcomes the objection to the normal (Gaus- 
sian) distribution function which implies the existence of particles of negative 

Another distribution function gaining popularity for characterizing crystal 
size distributions is the gamma function, expressed as 

f(d) = d x exp (dx/y)T(x + 1) (y/x) x+l (2.78) 

The parameters x and y, which give measures of the 'skewness' and 'size' of the 
distribution respectively, can be related to the crystallization process. The 
median size MS (= d) and standard deviation, a, may be calculated from 

d = y(x + l)/x and a = y{x + 1) ' jx 

Therefore from equation 2.75 the coefficient of variation is given by 

CV= 100(x+ ir 1/2 (2.79) 

Some skew distributions, particularly those of comminuted materials can be 
fitted by the Rosin-Rammler-Sperling (RRS) function. This relationship, 
based on one originally derived from probability considerations, may be 

P=l00exp[-(d/d')"] (2.80) 

where P = cumulative percentage oversize, d = particle size, and d' is a statist- 
ical mean size corresponding to P = 36.8% (100/e, where e = 2.718, the base of 
natural logarithms). Equation 2.80 indicates a linear relationship between 

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Physical and thermal properties 85 

log log(100// > ) and log d. The slope of the line, n, has been called a 'uniformity 
factor', i.e. « = tan#, where 9 is the angle between the RRS line and the 
horizontal. As the size distribution narrows towards a mono-sized dispersion, 
n — > oo; as it broadens, n —> 0. 

Special double logarithmic graph paper suitable for this type of plotting is 
available and the data from Table 2.14 are shown on such a plot in Figure 2.18b. 
From this plot it is possible to determine the 16, 50 and 84% cumulative 
percentages needed to calculate the MS/CV, as described above. Alternatively, 
the distribution can be characterized by the uniformity factor, n. In the example 
shown the median size c/50% = 870 um (the same as determined in Figure 2.18a), 
the statistical mean size a" = 1000 um, and the uniformity factor n = 1.8. 

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3 Solutions and solubility 

3.1 Solutions and melts 

A solution (gaseous, liquid or solid) is a homogeneous mixture of two or more 
substances. The constituents of liquid solutions are frequently called solvents 
and solutes, but despite common usage there is no fundamental reason why any 
one particular component of a solution should be termed the solvent, and con- 
siderable confusion can arise from adhering to rigid definitions. For example, 
a salt such as potassium nitrate fuses in the presence of small amounts of water 
at a much lower temperature than the pure salt does. The use of the term 
'solvent' for water would hardly seem to be justified in this case, and although it 
may seem strange to refer to 'a solution of water in potassium nitrate', this 
would be an equally acceptable description. Fusion is nothing more than an 
extreme case of liquefaction by solution, so it may be said that when a salt 
dissolves in water the salt, in fact, melts. 

Owing to the widespread and often indiscriminate use of the word 'melt', it 
is difficult to give a precise definition of the term. Strictly speaking, a melt is 
a liquid close to its freezing point, but in its general application the term also 
includes homogeneous liquid mixtures of two or more substances that would 
individually solidify on cooling to ambient temperatures. Thus a-naphthol 
heated above its melting point (96 °C) would be regarded as a melt, and so 
would a liquid mixture of a-naphthol and /3-naphthol (m.p. 122 °C). On the 
other hand, a liquid mixture of a-naphthol and methanol would normally be 
classified as a solution. However, no rigid definition is possible. The 
KNO3-H2O system quoted above, and the many well-known cases of 
hydrated salts dissolving in their own water of crystallization at elevated temper- 
atures, would in all probability be considered to be melts. 

3.2 Solvent selection 

Water is almost exclusively used as the solvent for the industrial crystallization 
of inorganic substances from solution. This fact is quite understandable 
because, apart from the relative ease with which a very large number of 
chemical compounds dissolve in it, water is readily available, cheap and innocu- 
ous. For these reasons water is used whenever possible even for the industrial 
crystallization of organic compounds, although for a variety of reasons other 
solvents may have to be used in this particular field. 

The selection of the 'best' solvent for a given crystallization operation is not 
always an easy matter. Many factors must be considered and some compromise 
must inevitably be made; several undesirable characteristics may have to be 
accepted to secure the aid of one important solvent property. There are several 

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Solutions and solubility 87 

hundred organic liquids that are potentially capable of acting as crystallization 
solvents, but outside the laboratory the list can be shortened to a few dozen 
selected from the following groups: acetic acid and its esters, lower alcohols and 
ketones, ethers, chlorinated hydrocarbons, benzene homologues, and light 
petroleum fractions. In many cases, of course, the solvent may already have 
been selected by the prevailing process conditions. In others, the cost of solvent 
recovery may override other considerations. 

A mixture of two or more solvents will occasionally be found to possess the 
best properties for a particular crystallization purpose. Common binary solvent 
mixtures that have proved useful include alcohols with water, ketones, ethers, 
chlorinated hydrocarbons or benzene homologues, etc. and normal alkanes 
with chlorinated hydrocarbons or aromatic hydrocarbons. 

A second liquid is sometimes added to a solution to reduce the solubility of 
the solute, cause its precipitation/crystallization and maximize the yield of 
product. It is necessary, of course, for the two liquids (the original solvent 
and the added precipitant) to be completely miscible with one another in all 
proportions. The process is commonly encountered, for instance, in the crystal- 
lization of organic substances from water-miscible organic solvents by the 
controlled addition of water. The term 'watering-out' is often used in this 
connection. This approach is also used to reduce the solubility of an inorganic 
salt in aqueous solution by the addition of a water-miscible organic solvent in 
salting-out precipitation processes (section 7.2.5). 

Some of the main points that should be considered when choosing a solvent 
for a crystallization process include the following. The solute to be crystallized 
should be readily soluble in the solvent. It should also be easily deposited from 
the solution in the desired crystalline form after cooling, evaporation, salting- 
out with an additive, etc. There are many exceptions to the frequently quoted 
rule that 'like dissolves like', but this rough empiricism can serve as a useful 
guide. Solvents may be classified as being polar or non-polar; the former 
description is given to liquids which have high dielectric constants, e.g. water, 
acids, alcohols, and the latter refers to liquids of low dielectric constant, e.g. 
aromatic hydrocarbons. A non-polar solute (e.g. anthracene) is usually more 
soluble in a non-polar solvent (e.g. benzene) than in a polar solvent (e.g. water). 
However, close chemical similarity between solute and solvent should be 
avoided, because their mutual solubility will in all probability be high, and 
crystallization may prove difficult or uneconomical. It should be noted that the 
crystal habit can often be changed by changing the solvent (section 6.4). 

Based on the nature of their intermolecular bonding interactions solvents 
may be conveniently divided into three main classes: 

1. polar protic, e.g. water, methanol, acetic acid; 

2. dipolar aprotic, e.g. nitrobenzene, acetonitrile, furfural; 

3. non-polar aprotic, e.g. hexane, benzene, ethyl ether. 

In polar protic solvents the solvent molecules interact by forming strong 
hydrogen bonds. In order to dissolve, the solute must break these bonds and 
replace them with bonds of similar strength. To have a reasonable solubility, 
therefore, the solute must be capable of forming hydrogen bonds, either 

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88 Crystallization 

because the solute itself is hydrogen bonded or because it is sufficiently basic to 
accept a donated hydrogen atom to form a hydrogen bond. If the solute is 
aprotic and not basic it cannot form strong bonds with the solvent molecules 
and therefore will have a very low solubility. 

In dipolar aprotic solvents, characterized by high dielectric constants, the 
solvent molecules interact by dipole-dipole interactions. If the solute is also 
dipolar and aprotic it can interact readily with the solvent molecules forming 
similar dipole-dipole interactions. If the solute is non-polar it cannot interact 
with the dipoles of the solvent molecules and so cannot dissolve. Protic solutes 
are found to be soluble in basic dipolar aprotic solvents because strong hydro- 
gen bonds are formed, replacing the hydrogen bonds between the solute mole- 
cules in the solid state. If a dipolar aprotic solvent is not basic, however, a 
protic solute will have a low solubility because the strong hydrogen bonds in 
the solid phase can only be replaced by weaker dipole-dipole interactions 
between solvent and solute molecules. 

In non-polar aprotic solvents, characterized by low dielectric constants, 
molecules interact by weak van der Waals forces. Non-polar solutes are readily 
soluble as the van der Waals forces between solute molecules in the solid phase 
are replaced by similar interactions with solvent molecules. Dipolar and polar 
protic solutes are generally found to have very low solubilities in these solvents 
except in cases where non-polar complexes are formed. 

Solvent power 

The 'power' of a solvent is usually expressed as the mass of solute that can be 
dissolved in a given mass of pure solvent at one specified temperature. Water, 
for example, is a more powerful solvent at 20 °C for calcium chloride than 
«-propanol (75 and 16g/100g solvent, respectively). At the same temperature 
M-propanol is a more powerful solvent than water for benzoic acid (42.5 and 
0.29g/100g solvent, respectively). 

The temperature coefficient of solubility is another important factor to be 
considered. For example, at 20 °C water is a more powerful solvent for potas- 
sium sulphate (11 g/lOOg water) than for potassium chlorate (7 g/100 g), but the 
converse is true at 80 °C (K 2 S0 4 , 21 g/100 g; KCIO3, 39 g/100 g). Thus, on cooling 
the respective saturated solutions from 80 to 20 °C, more than 80% of the 
dissolved KCIO3 would be deposited compared with less than 50% of the 
K 2 S0 4 . 

Both the solvent power and the temperature coefficient of solubility must be 
considered when choosing a solvent for a cooling crystallization process; 
the former quantity influences the volume of the crystallizer, and the latter 
determines the crystal yield. It frequently happens, especially in aqueous 
organic systems, that a low solubility is combined with a high temperature 
coefficient of solubility. For example, the solubilities of salicylic acid in water 
at 20 and 80 °C are 0.20 and 2.26 g/100 g, respectively. Therefore, on cooling 
from 80 to 20 °C, most of the dissolved solute (91%) is deposited, and conse- 
quently the solute yield is high. However, on account of the low solubility, even 
at the higher temperature, an excessively large crystallizer would be required to 

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Solutions and solubility 89 

give a reasonable production rate and consequently water could not be con- 
sidered as a suitable solvent for salicylic acid crystallization. 

Potassium chromate is an example of a solute with a reasonably high solu- 
bility in water and a low temperature coefficient of solubility (61.7 and 72.1 g/ 
100 g at 20 and 80 °C, respectively). The low yield on cooling (about 14 per 
cent) makes it necessary to effect crystallization in some other manner, such as 
a combination of cooling and evaporation, thus increasing the cost of the 
operation. An algorithm for the prediction of an optimal solvent or solvent 
mixture for cooling crystallization has been proposed by Nass (1994). 

Solvent hazards 

A few words on the subjects of purity and hazards might not be entirely out of 
place at this point, because their consideration is inevitable when choosing 
a solvent for a crystallization process. No deleterious impurity, dissolved or 
suspended, should be introduced into a crystallizing system. The solvent, there- 
fore, should be as clean and as pure as possible. No colouring matter should be 
permitted to affect the appearance of the final crystals. No residual odours 
should remain in the product after drying, a problem often encountered after 
crystallization from organic solvents with distinctive odours. If no previous 
experience has been obtained with a potential solvent, simple laboratory trials 
should be made, but due caution should be exercised in interpreting the results 
because laboratory filtration, washing and drying techniques generally prove to 
be much more efficient than the corresponding large-scale operations. 

The solvent should be stable under all foreseeable operating conditions: it 
should neither decompose nor oxidize, and it must not attack any of the 
materials of construction of the plant. When organic solvents are being used, 
care must be taken in choosing the correct gasket materials; most common 
types of rubber and many synthetic elastomers, for example, swell and disin- 
tegrate after prolonged contact with chlorinated hydrocarbons. 

The solute and solvent should not be capable of reacting together chemically, 
although solvate formation may be permitted under certain circumstances. 
Hydrated crystals are frequently desired as end-products, but should the anhyd- 
rous substance be required the necessary drying process may prove difficult and 
expensive. Methanol, ethanol, benzene and acetic acid are also known to form 
solvates with certain substances, and the loss of solvent on drying imposes an 
additional cost on the process. 

Highly viscous solvents are not usually conducive to efficient crystallization, 
filtration and washing operations. In general, therefore, solvents of low viscos- 
ity are preferred. If the solvent recovery process involves distillation, a reason- 
ably volatile solvent is desirable. On the other hand, the loss of a solvent with 
a high vapour pressure from filters and other processing equipment can be 
considerable and may prove both costly and hazardous. Solvents with freezing 
points above about — 5°C present wintertime storage and transportation diffi- 
culties. Benzene (f.p. 5°C) and acetic acid (f.p. 17 °C) are good examples. 

Most organic solvents employed in cystallization processes are flammable, 
and their use necessitates stringent operating conditions. Two of the most 

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90 Crystallization 

important properties of a flammable solvent are the flash point and explosive 
limits; the former is the temperature at which the mixture of air and vapour 
above the liquid can be ignited by means of a spark, and the latter refers to the 
percentages by volume in air between which the vapour mixture will, if ignited, 
explode in a confined space. Diethyl ether is an example of a solvent with a very 
low flash-point (— 30 °C) and wide range of explosive limits (1 to 50%). 

All organic solvents are toxic to a greater or lesser degree: the prolonged 
inhalation of almost any vapour will produce some harmful effect on a human 
being. Some solvents are acute poisons, some have a cumulative poisoning 
effect, and others produce narcosis or intoxication on inhalation, or dermatitis 
on contact with the skin. Information on these aspects and on the maximum 
vapour concentrations permitted in working areas can be obtained from spe- 
cialized reference books. The handbooks by Sax (1992) and Bretherick (1999) 
deal comprehensively with solvent properties and hazards. Health risks in the 
use of common solvents are dealt with in RSC/CEC (1986, 1988). 

3.3 Expression of solution composition 

The composition of a solution, or melt, may be expressed in many different 
ways, e.g. mass per unit mass of solvent, mass per unit mass of solution, mass 
per unit volume of solvent, and so on. The mass unit may refer to the dissolved 
species itself or to a solvated form, e.g. a hydrate. 

For the expression of crystallization kinetics there is some theoretical justi- 
fication for recording compositions on a molar basis, e.g. as kmolm~ 3 (i.e. 
molL~'), while mole fractions are most frequently used for thermodynamic 
calculations. Mass fractions are commonly used in the construction of phase 
diagrams, although the use of mole fractions is recommended for the repres- 
entation of reciprocal salt pair systems (section 4.7.2). 

For the purpose of expressing a mass balance on an item of process plant, 
there is considerable merit in expressing solution composition as mass of 
unsolvated solute per unit mass of solvent, particularly when temperature 
changes are expected. This avoids the need for further calculation to account 
for density changes. 

In view of the frequent need to make interconversions of composition units 
it is recommended that, whenever solution concentration measurements are 
made, the density of the solution at the relevant temperature is also measured 
and recorded (see section 3.9). 

Many of the above methods of solubility expression can lead to the use of the 
potentially misleading term 'percentage concentration'. For instance, an 
expression such as 'a 10 per cent aqueous solution of sodium sulphate' could 
be taken to mean, without further definition, any one of the following: 

10 g of Na 2 S0 4 in 100 g of water 

10 g of Na 2 SC>4 in 100 g of solution 

10 g of Na 2 S0 4 • 10H 2 O in 100 g of water 

10 g of Na 2 S0 4 • 10H 2 O in 100 g of solution 

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Solutions and solubility 91 

If 10 g of anhydrous Na 2 S0 4 in 100 g of water were the intended description 
of the solution concentration, this would then be equivalent to 

9.1gofNa 2 S0 4 in 100 g of solution 

20.6 g of Na 2 S0 4 • 10H 2 O in 100 g of solution 

26.0 g of Na 2 S0 4 • 10H 2 O in 100 g of water 

which gives some measure of the magnitude of the possible misinterpretation. To 
make matters even worse, the term 'percentage concentration' is often applied 
on a volume basis, e.g. 10 g of Na 2 S0 4 in 100 mL of water, of solution, and so on. 
Table 3.1 lists the interconversions between a number of the common expres- 
sions of solution composition. For convenience, the expressions are drafted in 
terms of aqueous systems, but the relationships are completely general if the 
terms 'unsolvated substance', 'solvate' and 'solvent' are substituted for 'anhy- 
drous substance', 'hydrate', and 'water', respectively. 

Table 3.1. Conversion factors for solution concentration units 


Equivalent expressions 


c 2 

R-C 3 

c 4 

C 5 
P - Ci 

c 6 

M A C 7 

R + (R - 1)C 4 

pR - C 6 

p-M A C 7 

c 2 

1 + Ci 



c 4 




M A C 7 

R(l + C 4 ) 


c 3 

1 + Ci 

RC 2 

c 4 

I + C4 




M H C 7 

c 4 


RC 2 
1 -RC 2 

C 3 

P - RC 5 

c 6 

p - c 6 


l-(R-l)C 2 

p- M H C 7 


1 + C, 



pC 4 
R(l + C 4 ) 

c 6 


M A C 7 

c 6 


1 + C, 

pRC 2 

pC 3 

pC 4 

1 + C 4 


M H C 7 

Ci and Cg 


pC 2 
M A 


pC 4 

M A 

c 6 

Ma(1 + Ci) 

M H (1 + C 4 ) 

c 2 

c 3 

c 4 



c 7 
c 8 



M A 

M H 

M w 


kg of anhydrous substance/kg of water 
: kg of anhydrous substance/kg of solution 

kg of hydrate/kg of solution 

kg of hydrate/kg of 'free' water 
: kg of anhydrous substance/m 3 of solution 

kg of hydrate/m 3 of solution 

kmol of anhydrous substance/m 3 of solution 

kmol of hydrate/m 3 of solution 
: mole fraction of anhydrous substance 

mole fraction of hydrate 

molar mass of anhydrous substance 

molar mass of hydrate 

molar mass of water 

: M H /M A 

density of solution (kgm~ 3 ) 

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92 Crystallization 

Interconversions between mass or molar units and those based on mole 
fractions are a little more complex than those in Table 3.1. The mole fraction 
x of a particular component in a mixture of several substances is given by 

= mi/ Mi 

mi/ Mi + m 2 /M 2 + m 3 /M 3 + ■ ■ ■ 

where m is the mass of a particular component, and M its molar mass. 

The relationship between compositions expressed in mole fractions and in 
other units is given by 

__ M W C| M W C 4 

9 " M w Ci +M A ~ Mn + (M H + M w - M A )C 4 


„ M w Ci M W C 4 

10 ~ M A - (M H - M A - M w )Ci ~ M W C 4 + M H ( ' ' 

A large number of terms have been used to express the relative solubility of 
a solute in a given solvent. The following, together with some examples of 
solubility (gL _1 ) at around ambient temperature, are the most frequently 

Practically insoluble 

BaS0 4 


Slightly soluble 

Ca(OH) 2 


Sparingly soluble 

PbCl 2 





Very soluble 



3.4 Solubility correlations 

In the majority of cases the solubility of a solute in a solvent increases with 
temperature, but there are a few well-known exceptions to this rule. Some 
typical solubilities for various salts in water are shown in Figure 3.1, where all 
concentrations are expressed as kg of anhydrous substance per 100 kg of water. 
In Figure 3.1a sodium chloride is a good example of a salt whose solubility 
increases only slightly with an increase in temperature, whereas sodium acetate 
shows a fairly rapid increase. 

The solubility characteristics of a solute in a given solvent have a consider- 
able influence on the choice of a method of crystallization. It would be useless, 
for instance, to cool a hot saturated solution of sodium chloride in the hope of 
depositing crystals in any quantity. A reasonable yield could only be achieved 
by removing some of the water by evaporation, and this is what is done in 
practice. On the other hand, a direct cooling crystallization operation would be 
adequate for a salt such as copper sulphate: cooling from 90 to 20 °C would 
produce about 44 kg of CUSO4 for every 100 kg of water present in the original 
solution. As the stable phase of copper sulphate at 20 °C is the pentahydrate the 

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« 40 r 





*■ / ?/ 



/ °/ n 






V / 

/ / / 


1 / 1 1 1/ ! Jl 1 1 1 


Solutions and solubility 93 

20 (- 

I / ' 1_M I 11 I I L 

20 40 60 80 100 

Concentration, kg/IOOkg water 


20 40 60 80 100 

Concentration, kg/IOOkg water 


Figure 3.1. Solubility curves for some salts in water: (a) smooth curves, (b) indicating 
occurrence of phase changes 

actual crystal yield would be about 69 kg of CUSO4 • 5H2O for every 100 kg of 
water present initially. 

Not all solubility curves are smooth, as can be seen in Figure 3.1b. A dis- 
continuity in the solubility curve denotes a phase change. For example, the 
solid phase deposited from an aqueous solution of sodium sulphate below 
32.4 °C will consist of the decahydrate, whereas the solid deposited above this 
temperature will consist of the anhydrous salt. The solubility of anhydrous 
sodium sulphate decreases with an increase in temperature. This negative 
solubility effect, or inverted solubility as it is sometimes called, is also exhibited 
by substances such as calcium sulphate (gypsum), calcium, barium and stron- 
tium acetates, calcium hydroxide, etc. These substances can cause trouble in 
certain types of crystallizer by causing a deposition of scale on heat-transfer 

The solubility curves for two different phases meet at the transition point, 
and a system may show a number of these points. For instance, three forms of 
ferrous sulphate may be deposited from aqueous solution depending upon the 
temperature: FeS0 4 • 7H 2 up to 56 °C, FeS0 4 • 4H 2 from 56 to 64 °C and 
FeS0 4 H 2 above 64 °C. 

The general trend of a solubility curve can be predicted from Le Chatelier's 
Principle which, for the present purpose, can be stated: when a system in 
equilibrium is subjected to a change in temperature or pressure, the system will 
adjust itself to a new equilibrium state in order to relieve the effect of the 
change. Most solutes dissolve in their near-saturated solutions with an absorp- 
tion of heat (endothermic heat of solution) and an increase in temperature 
results in an increase in the solubility. An inverted solubility effect occurs when 
the solute dissolves in its near-saturated solution with an evolution of heat 
(exothermic heat of solution). 

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94 Crystallization 

Strictly speaking, solubility is also a function of pressure, but the effect is 
generally negligible in the systems normally encountered in crystallization from 
solution. In the purification of melt systems, however, pressure manipulation 
can be utilized for separating organic isomers (section 7.3.2). 

Many equations have been proposed for the correlation and prediction of 
solubility data. Some are better than others, but none has been found to be of 
general applicability. In any case, an experimentally determined solubility is 
undoubtedly preferred to an estimated value, particularly in systems that may 
contain impurities. Nevertheless, there is frequently a need for a simple math- 
ematical expression of solubility to assist the recording and correlation of data. 

One of the most commonly used expressions of the influence of temperature 
on solubility is the polynomial 

c = A + Bt + Ct 2 + --- (3.4) 

where t is the temperature, e.g. in °C, and c is the solution composition, 
expressed in any convenient units. A, B, C, etc. are constants that depend on 
the units used. There is rarely any need to resort to higher-order polynomials 
for this empirical relationship. 

In addition to equation 3.4, a number of semi-empirical equations have been 
proposed for solubility correlation purposes, some of which are based on 
thermodynamic relationships relating to phase equilibria. Examples of some 
of the expressions that have found favour, at one time or another, are 

logx = A+Br (3.5) 

logx = A+BT+Cr 2 (3.6) 

logx = A+BJ- 1 (3.7) 

logx = A+BJ-' + CJ- 2 (3.8) 

logx = A+BT-' + ClogT (3.9) 

In all these relationships the solution composition x is expressed as mole 
fraction of solute and the temperature T is expressed in kelvins (K). The 
constants A, B and C in equations 3.4 to 3.9, of course, are not related to one 

Broul, Nyvlt and Sohnel (1981) came to the conclusion that, when tested 
against solubility data from 70 inorganic salts in water, the accuracy of the two- 
constant equations was consistently lower than that of the three-constant 
equations. However, they found that there was little to choose between indi- 
vidual equations in these two groups, but they did select equation 3.9 as being 
the most reliable. 

There is still considerable merit, however, in favour of equation 3.7 because 
of its simple form and its usefulness in the graphical estimation of transition 
points. Conventional solubility plots, such as those shown in Figure 3.1, can 
prove unreliable for this purpose when only a few data points are available, 
especially when the points lie on one or more different curves. Solubilities 
plotted in accordance with equation 3.7 are shown in Figure 3.2. Mole fraction 
concentrations x are recorded on the logarithmic abscissa and values of 10 3 T~ x 

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Solutions and solubility 









0.02 0.03 004 0.1 °-2 04 

Mole fraction anhydrous salt in water 
(logarithmic scale) 

Figure 3.2. Alternative method for the graphical representation of solubility data 

(because T~ x in the range 273-373 K is a very small quantity) are recorded on 
the right-hand linear ordinate scale. Alternatively, log-reciprocal graph paper 
can be used on which the temperature in degrees Celcius can be plotted directly, 
as on the left-hand ordinate in Figure 3.2. 

In Figure 3.1 the solubility of CUSO4 over the temperature range 0-100 °C is 
represented by a smooth curve, and the solubilities of Na2SC>4 and Na2CrC>4 are 
represented by smooth curves that intersect at transition points. Several advant- 
ages of the logx versus T~ x plot shown in Figure 3.2 immediately become 
apparent. For example, the data for the above three salts lie on a series of 
straight lines, which greatly assists interpolation and allows transition points 
to be identified with some precision. It is easier, for example, to produce the 
two straight lines for sodium sulphate in Figure 3.2, to meet at 32.4 °C than it 
is to extend the two corresponding curves in Figure 3.L The two straight lines 
for CUSO4 intersect at about 67 °C, which indicates a phase transition at 
this temperature; this transition between two different crystalline forms of 
the pentahydrate is not detected in Figure 3.L Incidentally, the transition 
CuS0 4 • 5H 2 ^ CuS0 4 • 3H 2 occurs at 95.9 °C. Only two of the transitions 
for the sodium chromate system are indicated in Figure 3.2. There are actu- 
ally three transition points in this system: IOH2O ^ 6H2O (19.6 °C), 
6H 2 ^ 4H 2 (26.6 °C) and 4H 2 ^ anhydrous (64.8 °C). 

The solubility data for sodium acetate are included in Figure 3.2 to illustrate 
the fact that straight lines do not always result from this method of plotting. 
Curved lines are often obtained for highly soluble substances, or in regions 
where the temperature coefficient of solubility is high, or in cases where several 
hydrates can exist over a narrow range of temperature. It is possible, of course, 
that the curved portion of the sodium acetate line in the region of about 
40-58 °C could be a series of straight lines representing hydrates other than 
the trihydrate, but there is no evidence to support this view. 

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96 Crystallization 

3.5 Theoretical crystal yield 

If the solubility data for a substance in a particular solvent are known, it is a 
simple matter to calculate the maximum yield of pure crystals that could be 
obtained by cooling or evaporating a given solution. The calculated yield will 
be a maximum, because the assumption has to be made that the final mother 
liquor in contact with the deposited crystals will be just saturated. Generally, 
some degree of supersaturation may be expected, but this cannot be estimated. 
The yield will refer only to the quantity of pure crystals deposited from the 
solution, but the actual yield of solid material may be slightly higher than that 
calculated, because crystal masses invariably retain some mother liquor even 
after filtration. When the crystals are dried they become coated with a layer of 
material that is frequently of a lower grade than that in the bulk of the crystals. 
Impure dry crystal masses produced commercially are very often the result of 
inadequate mother liquor removal. 

Washing on a filter helps to reduce the amount of mother liquor retained by 
a mass of crystals, but there is always the danger of reducing the final yield by 
dissolution during the washing operation. If the crystals are readily soluble in 
the working solvent, another liquid in which the substance is relatively insol- 
uble may be used. Alternatively, a wash consisting of a cold, near-saturated 
solution of the pure substance in the working solvent may be employed. The 
efficiency of washing depends largely on the shape and size of the crystals (see 
section 8.6.1). 

The calculation of the yield for the case of crystallization by cooling is quite 
straightforward if the initial concentration and the solubility of the substance at 
the lower temperature are known. The calculation can be complicated slightly 
if some of the solvent is lost, deliberately or accidentally, during the cooling 
process, or if the substance itself removes some of the solvent, e.g. by taking up 
water of crystallization. All these possibilities are taken into account in the 
following equations, which may be used to calculate the maximum yields of 
pure crystals under a variety of conditions. 

Let C\ = initial solution concentration (kg anhydrous salt/kg solvent) 
C2 = final solution concentration (kg anhydrous salt/kg solvent) 
W = initial mass of solvent (kg) 

V = solvent lost by evaporation (kg per kg of original solvent) 
R = ratio of molar masses of hydrate and anhydrous salt 

Y = crystal yield (kg) 

Substance crystallizes unsohated {e.g. anhydrous salt) 

Total loss of solvent: Y = WC\ (3.10) 

No loss of solvent: Y = W{C\ - C 2 ) (3.11) 

Partial loss of solvent: Y = W[C\ - C 2 (l - V)] (3.12) 

Substance crystallizes as a solvate 

Total loss of free solvent: Y = WRC\ (3.13) 

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Solutions and solubility 97 

No loss of solvent: Y = WR( ~ Cl Z Cl) (3.14) 

1 - C 2 (R - 1) 

p fii fit v WR ^ ~ C 2(! ~ V)] ni « 

Partial loss ol solvent: Y = — — — (3.15) 

1 - C 2 (R - 1) 
Equation 3.15 can, of course, be used as the general equation for all cases. 

Example 1 

Calculate the theoretical yield of pure crystals that could be obtained from 
a solution containing 100 kg of sodium sulphate (mol.wt. = 142) in 500 kg of 
water by cooling to 10 °C. The solubility of sodium sulphate at 10 °C is 9 kg 
of anhydrous salt per 100 kg of water, and the deposited crystals will consist of 
the decahydrate (mol. wt. = 322). Assume that 2 per cent of the water will be 
lost by evaporation during the cooling process. 

R = 322/142 = 2.27 
C\ = 0.2 kg Na2S04 per kg of water 
C 2 = 0.09 kg Na 2 S0 4 per kg of water 
W = 500 kg of water 

V = 0.02 kg per kg of water present initially 

Substituting these values in equation 3.15 gives 

_ 500 x 2.27[0.2 - 0.09(1 - 0.02)] 
Y ~ 1 - 0.09(2.27 - 1) 

Yield = 143 kg Na 2 S0 4 • 10H 2 O 

To determine the crystal yield from a vacuum crystallizer (section 7.5.3) it is 
necessary to estimate the amount of solvent evaporated, V. This depends on the 
heat made available during the operation of the crystallizer, i.e. the sum of 
the sensible heat drop of the solution, which cools from the feed temperature to 
the equilibrium temperature in the vessel, and the heat of crystallization liber- 
ated. The heat balance, therefore, will be 

VW\ = c(h - t 2 )W(\ + Ci) + A c r (3.16) 

where, in addition to the symbols defined for equation 3.15, 

A v = enthalpy of vaporization of solvent (kJkg -1 ) 

A c = heat of crystallization of product (kJkg -1 ) 
t\ = initial temperature of solution (°C) 
Z 2 = final temperature of solution (°C) 
c = mean specific heat capacity of solution (kJkg -1 K~') 

Substituting for the value of Y from equation 3.15 and simplifying 

= Ac*(Ci - C 2 ) + cjh - t 2 )(\ + CQ[1 - C 2 (R - 1)] 
A v [l - C 2 (R - 1)] - X C RC 2 

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Example 2 

Estimate the yield of sodium acetate crystals (CH^COONa • 3H2O) from a 
continuous vacuum crystallizer operating with an internal pressure of 15mbar 
when it is supplied with 2000 kg h~ of a 40 per cent aqueous solution of sodium 
acetate (0.4 kg of anhydrous salt in 0.6 kg water) at 80 °C. The boiling point 
elevation of the solution may be taken as 11.5°C. 

Heat of crystallization of CH 3 COONa • 3H 2 0, A c = 144kJkg~' 

Specific heat capacity of the solution, c = 3.5kJkg~' K _1 

Latent heat of vaporization of water at 15mbar, A v = 2.46 M J kg -1 

Boiling point of water at 15mbar = 17.5 °C 

Operating temperature =17.5°+11.5°C =29°C 

Solubility at 29 °C, C 2 = 0.539 kg kg" 1 

Initial concentration, C\ = 0.4/0.6 = 0.667 kg kg - 

Initial mass of water in feed, (f = 0.6x 2000 = 1200 kg h" 1 

Ratio of molar masses, R = 136/82 = 1.66 

The quantity of water vaporized is calculated from equation 3.17: V = 
0.1 53 kg/kg of water present originally which, when substituted in equation 
3.15, gives the crystal yield as Y = 660kgh~ of sodium acetate trihydrate. 

3.6 Ideal and non-ideal solutions 

An ideal solution is one in which the interaction between solute and solvent 
molecules is identical with that between the solute molecules and the solvent 
molecules themselves. From this definition alone it is clear that a truly ideal 
solution is most unlikely to exist, but the concept is still very useful as a 
reference condition. For instance, if the solute and solvent did form an ideal 
solution, the solubility could be predicted from the van't Hoff equation: 

AH f 



1 1 

7>~ T 


where x is the mole fraction of the solute in the solution, T is the solution 
temperature (K), 7f is the fusion temperature (melting point) of the solute (K), 
A//f is the molal enthalpy of fusion of the solute (J mol~ ) and R is the gas 
constant (8.314 Jmor'K -1 ). 

For example, the solubility of naphthalene at 20 °C in an ideal solution may 
be calculated from its melting point (80 °C) and enthalpy of fusion 
(18.8 kJ moP ) to give x = 0.269. In principle, therefore, by performing such 
calculations over a range of temperatures, an 'ideal' solubility curve may be 
constructed, but it is important to note that any such calculated solubility is 
expressed without reference to any particular solvent. Furthermore, the 
assumption of ideality for most real solutions is generally unjustified. 

The potential unreliability of equation 3.18 in predicting solubility can be 
demonstrated by comparing the above calculated ideal solubility (x = 0.269) of 
naphthalene with measured solubilities in a few common solvents: 

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Solutions and solubility 99 

benzene 0.241 toluene 0.224 CC1 4 0.205 hexane 0.090 

Even in the case of benzene, with its chemical similarity to naphthalene, 
ideality is barely approached (the predicted solubility is some 12% too high). 
The solution in hexane, however, is highly non-ideal (the prediction is 200% 
too high). 

The van't Hoff equation can also be written in the form: 

m*=-^ + ^ (3.18a) 

RT R v ' 

since A//f = 7f AS[. ASf is the molal entropy of fusion. It should be under- 
stood, however, that even though a plot of In x versus T -1 may give a straight 
line, its slope may differ from — AH{/R if the solution exhibits non-ideal 
behaviour. In such cases, the enthalpy and entropy of mixing must be taken 
into account by replacing AH[ with AH^ (of dissolution) and AS[ by ASd 
(Beiny and Mullin, 1987), i.e., using: 

. A// d AS d „ 1ftM 

\nx = (3.18b) 

RT R V ' 

Another approach stems from a consideration of the Gibbs free energy 
change AG for a dissolution process, which in general may be expressed in 
terms of the enthalpy and entropy changes associated with the mixing process: 

AG = AH -T AS (3.19) 

For the formation of an ideal solution, e.g. by mixing two liquids, the Gibbs 
free energy may also be expressed as 

AG = RTlnx (3.20) 

where x is the solution composition expressed as a mole fraction of one of the 
components. The entropy change accompanying this dissolution process, from 
equation 3.19, is 

A5=-Rlnx (3.21) 

since the enthalpy of mixing, AH, is zero for an ideal solution. 

The overall free energy change may also be expressed in terms of the activity, 
a, of one of the components: 

AG = RT\na (3.22) 

In other words, if the solution is ideal, 

a = x (3.23) 

For non-ideal solutions, however, equation 3.23 has to be modified by the 
appropriate activity coefficient, 7, i.e. 

a = 7X (3.24) 

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100 Crystallization 

Similarly, equation 3.18 should be expressed as 

1 1" 
Y ( ~ T 

In (xi) = —^- 


The key to the use of equation 3.25 for the prediction of solubilities in non-ideal 
systems is a reliable estimation of the activity coefficient 7. For organic 
solutes in organic solvents, this may be achieved (Gmehling, Anderson and 
Prausnitz, 1978) by the UNIFAC group contribution method which is 
discussed, together with other techniques for the prediction of solubility data, 
in section 3.10. 

3.6.1 Activity and ionic strength 

The colligative properties of solutions, e.g. osmotic pressure, boiling point 
elevations, freezing point depression and vapour pressure reduction, depend 
on the effect of solute concentration on the solvent activity. The chemical 
potential, \x, of a non-electrolyte in dilute solution may be expressed by 

M = Moc + R7 , lnc (3.26) 

where c is the solute molar concentration (molL~ ) and /j, oc is the standard 
chemical potential also expressed on a molar basis (J moP ). 

Equation 3.26 is unsatisfactory for non-electrolyte solute concentrations in 
excess of about 0.1 molar, and it cannot be applied to solutions of electrolytes 
for concentrations greater than about 10~ 3 molar. For such cases an expression 
based on activity rather than concentration should be applied, e.g. 

M = Moc + Rrin<3 c (3.27) 

where a c is the solute activity expressed on a molar basis, which is related to 
composition c through the corresponding activity coefficient, j c , by 

a c = c'7 c (3.28) 

The activity coefficient becomes unity at infinite dilution, i.e. when ideality may 
be assumed. 

For electrolyte solutions it is more appropriate to use the mean ionic activity, 
a±, defined with respect to the mean ionic concentration and mean ionic 
activity coefficient by 

a c = a v ±c = (c ±7±c r = (Qn ±c f (3.29) 

where v is the number of moles of ions in 1 mole of electrolyte, i.e. 

v = v+ + v _ (3.30) 


Q = {v v ^v v -f v (3.31) 

For non-electrolytes, v = 1. 

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Solutions and solubility 


From the Debye-Hiickel theory of electrolytes, the limiting (infinite dilution) 
law gives the mean activity coefficient of the ion as 

\og 1± = -A\z + z_\I i ' 2 
where the ionic strength, /, expressed as mol L , is defined by 

l = \ 

J2 ctz 



where c r - is the concentration (mol L ) of the z'th ionic species and z,- the 
valency, and z + and z_ are the valencies of the cation and anion, respectively. 
The Debye-Hiickel constant, A, has values of 0.493, 0.499, 0.509 and 0.519 at 
5, 15, 25 and 35 °C, respectively. 

Similar relationships to those of equations 3.26 to 3.33 may be written for 
solution compositions expressed as molality, m (mol/kg of solvent), and 
mole fraction, x, respectively. For example, equation 3.26 could be written as 

M = fom + RT\nm 






For simplicity, however, only the molar-based ionic strength (/ = I c ) will be 
used subsequently in this section. 

The Debye-Hiickel limiting law (equation 3.32) has to be modified for all but 
the most dilute solutions, and many modifications have been proposed. For 
example, the Giintelberg equation: 

M = 

= /i ox + RTlnx 


equation 3.33 as either 


= £"»/*? 






/ + 7 1 /' 2 


is useful for solutions of sparingly soluble electrolytes, and the Davies 







is generally quite satisfactory for values of I up to about 0.2 mol L (Davies, 
1962; Nancollas, 1966). For concentrated mixed electrolyte solutions, more 
complex relationships have to be employed (see section 3.10). 

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102 Crystallization 

The calculation of the ionic strength of a mixed solution of electrolytes may 
be illustrated, using equation 3.33, for a mixture of equal volumes of 0.1 molL~ 
NaCl and CaCh solutions, assuming complete dissociation: 

/ = i([(Na+) x l 2 ] + [(Ca 2+ ) x 2 2 ] + [(CI") x l 2 ]) 

= £(0.1 x 1 + 0.1 x 4+3 x 0.1 x 1) 

= 0.4molL _1 

3.6.2 Association and dissociation 

For incompletely dissociated electrolytes, it is generally convenient to define 
another activity coefficient, the mean activity coefficient of ions in solution, j' ± . 
For a binary electrolyte which dissociates according to 

M v+ A v _ =; v+M z+ + v-A z ~ (3.36) 

the concentration of free ions in a solution of molar concentration, c, is 
ac, where a is the degree of dissociation, and the activity coefficients are 
related by 

C7±c = acj'± c (3.37) 


a = 7±c /7ic (3-38) 

The degree of dissociation, a, of dissolved electrolyte was first expressed by 
Arrhenius in 1887 as the ratio of the molar conductivity (see section 3.6.3) of 
the solution, A, to that of a solution at its most extreme dilution, Aq, i.e. 

a = A/A (3.39) 

The degree of dissociation can also be expressed in terms of the van't Hoff 
factor, i, and the number of moles of ions, v, in one mole of solute: 

i=\-a + va (3.40) 


a = ^- (3.41) 

v — \ 

For strong electrolytes (virtually complete dissociation, a — ► 1) 

i = v (3.42) 

The dissociation of an electrolyte molecule in solution into oppositely 
charged ions, however, is by no means a simple matter. The ionic association 
theory, first developed by Bjerrum in 1926, indicates that some kind of associa- 
tion will still exist between oppositely charged ions even when they are several 
molecular diameters apart. The rates of dissociation and reformation, of mole- 
cules or other complexes, are extremely fast and it is doubtful if the ions can 

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Solutions and solubility 103 

ever truly be considered to be 'free'. Ion association should be taken into 
account in any complete treatment of aqueous electrolyte solutions since it 
reduces the value of the ionic activity coefficients. 

3.6.3 Conductance and conductivity 

The current flowing through an electrolyte solution is proportional to the 
reciprocal of its resistance, R (Ohm's law). This quantity, \jR, is called the 
conductance of the solution, which formerly was expressed as reciprocal ohms 
(mhos, tt~ l ), but now bears the SI unit name of the Siemens, S. Thus, a solution 
with a resistance of 10 Q has a conductance of 0.1 S. 

The resistances of different solutions can be compared through the quantity 
known as the resistivity, p, defined by the relationship 

p = Ra/l (3.43) 

where / is the length of the conductivity path in the solution and a is its cross- 
sectional area. The units of p are the ohm metre, Q m, or in SI units, Siemens - 
metre, S -1 m. 

In a similar manner to resistances, the conductances can be compared 
through the quantity known as the conductivity, k (fi _1 m _1 or Sm~ ), which 
is the reciprocal of the resistivity, i.e. 

k = - (3.44) 


For electrolyte solutions the molar conductivity, A (the conductivity per 
mole of electrolyte), is a useful characteristic since it allows comparisons to 
be made between solutions of different substances. A is defined by 

A = k/c (3.45) 

where c is the solution concentration, expressed as molm~ 3 , giving A the units 
of Q~ l m 2 moP , i.e. Sm 2 moF . According to Kohlrauch's law, the value of A 
at infinite dilution, Ao, is the sum of the corresponding molar ionic conductiv- 
ities, also at infinite dilution (see Tables 3.2 and A. 13), i.e. 

Ao = A+ + A (3.46) 

The utility of the above relationships in estimating the solubility of sparingly 
soluble salts is discussed in section 3.9.3. 

Furthermore, in addition to its straightforward application, equation 3.46 
can be used to calculate Ao for weak electrolytes, e.g. organic acids, from Aq 
values for their strong electrolyte salts. For example, Ao for acetic acid can be 
calculated (Moore, 1972) from values of sodium acetate, HC1 and sodium 
chloride. At 25 °C: 

Ao(HAc) = Ao(NaAc) + A (HC1) - A (NaCl) 
= (91+425- 128) x 10 -4 
= 388 x 10 -4 Sm 2 mol -1 

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104 Crystallization 

Table 3.2. Some molar ionic conductivities at infinite dilution at 25 °C 


10 4 A+ 



(Sm 2 mor') 

(Sm 2 mor') 













Ag + 












±Mg 2 + 


cio 4 


iCa 2 + 


CH 3 CO^ 


iSr 2+ 


ico 2 - 


iBa 2 + 


±so 2 - 


ipb 2+ 


|Cr0 2 - 


The values of Ao expressed here as x 10 4 S m 2 mol ' are equivalent to the former common units of 
Q~ l cm 2 mor'. 

3.6.4 Solubility products 

The solubility of a sparingly soluble electrolyte in water is often expressed in 
terms of the concentration solubility product, K c . To take the simplest case, if 
one molecule of such an electrolyte dissociates in solution into x cations and y 
anions according to the equation 

M x A y ^ xM z+ + yA z - (3.47) 

where z + and z~ are the valencies of the ions, then for a saturated solution 

{c+fic-f = constant = K c (3.48) 

where c+ and c_ are the ionic concentrations expressed as mol L~ . 

Solubility products are often recorded, for convenience, as pK values where 
pK c =—\ogK c . Thus a value of K c = 3.9 x 1CT 6 would be reported as 
pK c = 5.4. 

For a salt which produces two ions per molecule (1-1, 2-2, etc. electrolytes, 
i.e. for x = y = 1) c + = C- = c* , where c* is the equilibrium solubility 
(molL" 1 ). 

Therefore, equation 3.48 becomes 

c* = (K c f 2 (3.49) 

In general 

c* = (K c /x x f) v(x+y) (3.50) 

Therefore, for a 2-1 electrolyte 

c* = (KJ4) 113 (3.51) 

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Solutions and solubility 105 

and for a 3-1 electrolyte 

c* = (K c /27) 114 (3.52) 

For example, the solubility products of silver bromide, lead iodide and 
aluminium hydroxide at 18 °C are 4.1 x 1(T 13 , 9.3 x 1(T 9 and 1.1 x 1CT 15 
respectively, so their solubilities in water at this temperature may be expressed as: 

AgBr (equation 3.49): 

c* = (4.1 x 1(T 13 ) 1/2 = 6.4 x 10 -7 molL -1 
PM2 (equation 3.51): 

c* = [(7.5 x 1(T 9 )/4] 1/3 = 1.2 x lO^molL" 1 
Al(OH) 3 (equation 3.52): 

c* = [(1.1 X 1(T 15 )/27] 1/4 = 8.0 x lO^molL" 1 

However, the simple solubility product principle has extremely limited use. It 
should, for example, be restricted to solutions of very sparingly soluble salts 
(< 10~ 3 molL~ ). For more concentrated solutions it is necessary to adopt a 
more fundamental approach involving the use of activity concepts. 
The activity solubility product, K a , is defined 

(a+Yia-f = constant = K a (3.53) 

where a + and a_ are the ionic activities. As the activity of an ion may be 
expressed in terms of the ionic concentration, c, and the ionic activity coeffi- 
cient, 7, equation 3.53 may be written 

(c + 7+)*(c-7-y = K a (3.54) 


K a = K C { 1± Y (3.55) 

where j± is the mean ionic activity coefficient and v (=x + y) is the number of 
moles of ions produced by one mole of electrolyte. 

So the concentration solubility product is equal to the activity solubility only 
when 7± = 1, i.e. at infinite dilution. In practice, K a and K c may be assumed 
approximately equal for concentrations up to about 10~ 3 molL~ , but above 
this concentration significant deviations can occur. The activity of an ion 
depends on the concentration of all the other ions in solution, so the presence 
of a dissolved foreign electrolyte can greatly influence the value of j± of a 
sparingly soluble salt. 

A number of cases which appear anomalous when the simple solubility 
product is used can be explained when activity coefficients are taken into 
account. For instance, the addition of a common ion generally decreases the 
solubility of a salt, but cases are known where additions of a salt with a 
common ion result in increase in solubility. The reason for this is that a large 

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Ionic strength 

Figure 3.3. Relative increase in solubility with increase in ionic strength: (a) BaSC>4 in 
KNO3 solution, (b) AgCl in KNO3 solution. (After Lewin, 1960) 

increase in ionic concentration can cause a reduction in the activity coefficients. 
Thus from equation 3.54 an increase in c_ will result in a decrease in c+, i.e. 
precipitation of the sparingly soluble salt, if 7+ and 7_ remain fairly constant, 
but an increase in c_ to a value which reduces both 7+ and 7_ must result in 
an increase in c + if K a is to remain constant. The addition of a salt without 
a common ion often increases the solubility; this again is the result of the 
increased ionic concentration reducing the activity coefficients (Figure 3.3). 

Lewin (1960) has pointed out that even equation 3.54 does not represent the 
true situation since a saturated aqueous solution in equilibrium with the solid 
phase involves the reversible reaction 

solid + water ^ saturated solution 

so at constant temperature and pressure 

M x A y + (xb + yc)H 2 ^ x(M z+ ■ bU 2 0) + y(A z - ■ cH 2 0) 

solid solvent saturated solution 

where b and c are the numbers of water molecules associated with the cation 
and anion respectively. Consequently, under equilibrium conditions, by the 
Law of Mass Action: 

(a+) x (a_) 

i r 

( fl solid)( a water) 




which Lewin called the comprehensive activity solubility product. The use of this 
thermodynamically rigorous form of the solubility product can resolve most 
apparently anomalous problems. 

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Solutions and solubility 107 

Common and diverse ion effects 

The addition of an electrolyte to a saturated solution of a sparingly soluble salt 
with a common ion depresses the solubility of the latter (the common ion effect) 
and leads to its precipitation. For example, the solubility product of AgCl at 
25 °C is 1.56 x 10~ 10 (K a = K c at this dilution), i.e. 

[Ag + ][Cr]= 1.56 x 1(T 10 

giving, from equation 3.49, a solubility of 

c* = 1.25 x 10 -5 molL -1 

If a small quantity of a more soluble chloride, e.g. NaCl, is added to a satur- 
ated solution of AgCl the CI - concentration will temporarily exceed 
1.25 x 10- 5 molL -1 , i.e. 

[Ag + ][CP]> 1.56 x 1(T 10 

This unstable condition cannot persist so the system readjusts itself until the 
new ionic product equals the solubility product and this results in the precipita- 
tion of some of the AgCl. 

The solubility product principle can only be strictly applied to equilibrium 
conditions, although it has often been used to explain such precipitations as 
those encountered in qualitative analysis by the traditional wet-test methods. 
However, these sudden precipitations do not take place under anything like 
equilibrium conditions and the fact that reasonably successful predictions can 
usually be made is mainly due to the enormous excess ionic concentrations 
(supersaturations) generated compared with those required by the corresponding 
solubility products. Errors of magnitude of 10 5 — 10 7 per cent have been estimated 
(Lewin, 1960) for such calculations and these clearly swamp other variations such 
as neglect of solute activity coefficients, complex ion formation, etc. 

Whereas the presence in solution of an ion in common with a sparingly 
soluble salt can significantly decrease the salt solubility, the presence of an 
ion not in common with any of those of the solute can increase the solute 
solubility on account of the increase in ionic strength. For example, the solu- 
bility of silver bromide in water is increased by around 30% in a 0.1 molL~ 
sodium nitrate aqueous solution, as can be seen in the following rough calcula- 

The solubility of silver bromide in water at 15 °C is about 6 x 10~ 7 mol L~ . 
At this low concentration the 'activity' and 'concentration' solubility products 
may be assumed to be equal (see equation 3.55), i.e. 

K c = (c*) 2 = 3.6 x 10~ 13 = K a 

For a 0.1 molar solution of sodium nitrate, using equations 3.35 with a value 
of A = 0.499 appropriate for 15 °C, the calculated ionic activity coefficient 7+ 
is 0.783, from which it may be estimated that 

K c = 3.6 x 10~ 13 /(0.783) 2 = 5.87 x 10~ 13 

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108 Crystallization 

giving the solubility 

c* = K]) 1 = 7.7 x lO^molL -1 

Temperature effects 

The ionic product of water 

K w = [H+][OH-] (3.57) 

is temperature dependent, rising from about 10~ 15 at °C, to 10~ 14 at 25 °C and 
10~ 13 at 60 °C. Such a large variation is of considerable significance in the 
precipitation of sparingly soluble metal hydroxides, and its neglect can lead to 
gross errors. Fe(III) hydroxide, for example, dissociates according to: 

Fe(OH) 3 ^ Fe 3+ + 30H 

and the solubility product 
K c = [Fe 3+ ][OH-] 3 


[Fe J+ ] = r = ^— 

[OH-] 3 [* w ] 3 

Thus at a given pH the Fe 3+ concentration above which precipitation may be 
considered possible is 

[Fe 3+ ] oc K- 3 

A value of K w = 10~ 14 is commonly used in rough calculations, but this is only 
correct at 25 °C. If it were to be used for conditions at 15 °C, where the correct 
value of K„ is 0.45 x 10~ 14 , an error of about (1/0.45) -3 x 100, i.e. > 1000 per 
cent, would be incurred. 

3.7 Particle size and solubility 

The relationship between particle size and solubility, originally derived for 
vapour pressures in liquid-vapour systems by Thomson (who became Lord 
Kelvin in 1892) in 1871, utilized later by Gibbs, and applied to solid-liquid 
systems by Ostwald (1900) and Freundlich (1926) may be expressed in the form 




2M ^ (3.58) 


where c(r) is the solubility of particles of size (radius) r, c* is the normal 
equilibrium solubility of the substance, R is the gas constant, T is absolute 
temperature, p is the density of the solid, M is the molar mass of the solid in 
solution and 7 is the interfacial tension of the solid in contact with the solution. 

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Solutions and solubility 109 

The quantity v represents the number of moles of ions formed from one mole of 
electrolyte (equation 3.30). For a non-electrolyte, v = 1. 

Confusingly, but for understandable reasons, equation 3.58 is referred to in 
the literature by a variety of names such as the Gibbs-Thomson, Gibbs-Kelvin 
and Ostwald-Freundlich relation. For consistency, however, the designation 
'Gibbs-Thomson' will be used throughout this work. 

As a result of the particle size solubility effect, solution compositions may 
exceed greatly the normal equilibrium saturation value if the excess solute 
particles dispersed in the solution are very small. For most solutes in water, 
however, the solubility increase only starts to become significant for particle 
sizes smaller than about 1 um. 

For example, for barium sulphate at 25 °C: T = 298 K, M = 233 kgkmol -1 , 
j, = 2, p = 4500kgm- 3 , 7 = 0.13Jm- 2 , R = 8.3 x 10 3 JkmoF 1 K _1 . Thus for 
a 1 urn, crystal (i*=5x 10~ 7 m), cjc* = 1.005 (i.e. 0.5% increase). For 0.1 urn, 
cjc* = 1.06 (i.e. 6% increase) and for 0.01 um, cjc* = 1.72 (i.e. 72% increase). 

For a very soluble organic compound such as sucrose (M = 342 kg kmol - , 
v=\, p= 1590kgm~ 3 , 7=10- 2 JnT 2 ) the effect is similar: 1 urn (0.4% 
increase), 0.1 um (4%), 0.01 um (40%). All such calculated values, however, 
should be treated with caution, not only because of the unreliability of 7 values 
but also because the Gibbs-Thomson effect may cease to be influential at 
extremely small crystal sizes (see Figure 3.4). 

For practical application to crystals, equation 3.58 could be more usefully 
redrafted in terms of particle size expressed as a convenient length parameter, 
L, coupled with an approximate overall shape factor, F: 





For spheres and cubes F = 6 (L = diameter or length of side). For other shapes 
F > 6, e.g. for an octahedron F = 7.35 (section 2.14.3). 

Strictly speaking, equation 3.58 should be expressed in terms of solution 
activities rather than concentrations (Enustiin and Turkevich, 1960). Further- 
more, it involves a number of assumptions that may not always be valid. For 
example, the solid-liquid interfacial tension (section 5.6) is implicitly assumed 
to be independent of particle size, and no account is taken of any ionization or 

Particle size /•—» 
Figure 3.4. The effect of particle size on solubility 

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110 Crystallization 

dissociation of the solute in solution. This latter effect, discussed in detail by 
Jones and Partington (1915), leads to the relationship 



2M 7 

n j. mit (3 ' 60) 

(1 — a + va)K.l pr 

where a is the degree of dissociation. For the case of complete dissociation 
(a = 1) equation 3.60 reduces to equation 3.58. 

Another fault of equation 3.58 is that it postulates an exponential increase in 
solubility to infinity with a reduction in particle size to zero: 

c(f) = c* exp (2 M 7/^R T pr) (3.61) 

To overcome this anomaly, Knapp (1922) considered that the total surface 
energy (interfacial tension) of very small solid particles should be regarded as 
the sum of their 'normal' surface energy plus the surface electrical charge that 
such particles would carry. From these considerations he derived for the case of 
isolated charged spheres an equation of the form 

c(r) = c*exp04r -1 - Br 4 ) (3.62) 

where A = 2jM/RTp and B = q 2 Mj^TTnRTp, q being the electrical charge on 
the particle of radius r and k its dielectric constant. Equation 3.62 gives a curve 
of the type shown in Figure 3.4. Subsequent work has tended to lend support to 
Knapp's postulation and measurements made by Harbury (1946) indicated 
maximum:equilibrium solubility ratios of 3, 6 and 13 for salts KNO3, KCIO3, 
K2Cr207 and Na2S04 • IOH2O in water. 

3.8 Effect of impurities on solubility 

So-called pure solutions are rarely encountered outside the analytical labor- 
atory, and even then the impurity levels are usually well within detectable limits. 
Industrial solutions, on the other hand, are almost invariably impure, by any 
definition of the term, and the impurities present can often have a considerable 
effect on the solubility characteristics of the main solute. 

If to a saturated binary solution of A (a solid solute) and B (a liquid solvent) 
a small amount of the third component C (also soluble in B) is added, one of 
four conditions can result. First, nothing may happen, although this is com- 
paratively rare, in which case the system remains in its original saturated state. 
Second, component C may react or otherwise combine or react chemically with 
A by forming a complex or compound, thus altering the whole nature of the 
system. In the third case, the presence of component C may make the solution 
super-saturated with respect to solute A, which would then be precipitated. In 
the fourth case, the solution may become unsaturated with respect to A. The 
terms 'salting-out' and 'salting-in' are commonly used to describe these last two 
cases, particularly when electrolytes are involved. 

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Solutions and solubility 111 

The salting-out effect of an electrolyte added to an aqueous solution of a 
non-electrolyte, can often be represented by the empirical equation 

log— = kC + (3 (3.63) 


where c* and c*' are the equilibrium saturation concentrations (mol L~ of the 
non-electrolyte in pure water and in a salt solution of concentration C 
(molL~ ), respectively. The constants k, called the salting parameter, and (3 
refer to one particular electrolyte and its effect on one particular non-electrolyte 
at a given temperature. This type of relationship, which often applies with a 
reasonable accuracy for low non-electrolyte concentrations and electrolyte 
concentrations up to 4 or 5 mol L~ , is commonly employed to characterize 
the precipitation of proteins from aqueous solution using inorganic electrolytes 
(Bell, Hoare and Dunnill, 1983). 

Occasionally the presence of an electrolyte increases the solubility of a non- 
electrolyte (negative value of k) and this salting-in effect is exhibited by several 
salts, with large anions and cations, which themselves are very soluble in water. 
Sodium benzoate and sodium />-toluenesulphonate are good examples of these 
hydrotropic salts and the phenomenon of salting-in is sometime referred to as 
hydrotropism. Values of the salting parameter for three salts applied to benzoic 
acid are NaCl (0.17), KC1 (0.14) and sodium benzoate (-0.22). Long and 
McDevit (1952) have made a comprehensive review of salting-in and salting- 
out phenomena. 

Halstead (1970) reported that potassium sulphate once dissolved would 
not recrystallize from aqueous solutions contaminated with traces of chro- 
mium(III) or iron(III) and suggested that these impurities prevented the nuclea- 
tion of K2SO4 crystals. Trace impurities can sometimes have highly unexpected 
effects on equilibrium solubility measurements. For example, the solubility of 
potassium sulphate is significantly lowered when measured by dissolving 
K2SO4 crystals in water containing ppm traces of Cr(III) (Kubota et al., 
1988). A measured value obtained under these circumstances, however, is only 
an apparent or pseudo-solubility, the value of which is determined by two 
competing rate processes, viz., the adsorption of Cr(III) species on the K2SO4 
crystals and the dissolution of the crystals. Interestingly, a false solubility 
measurement is also obtained when, instead of approaching equilibrium from 
the undersaturated state, it is approached from the supersaturated condition, 
e.g., by cooling a solution to deposit excess solute. In this case, the recorded 
pseudo-solubility is higher than the true equilibrium value because the Cr(III) 
species in solution adsorb on the depositing K2SO4 crystals, retard their growth 
and prevent complete desupersaturation of the solution. The actual species 
adsorbed, one of the many possible hydroxo-Cr(III) complexes, depends on 
the particular salt added as impurity and the pH of the solution (Kubota et al., 
1994). The magnitude of the decrease in the measured pseudo-solubilities 
depends on pH, impurity concentration and the particular salt of the impurity 
added. Similar patterns of behaviour are seen when traces of Fe(III) salts are used 
as the added impurity (Kubota et al, 1999). An account of the effects of trace 
impurities on crystal growth and dissolution processes is given in section 6.2.8. 

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112 Crystallization 

When considerable quantities of a soluble impurity are present in, or delib- 
erately added to, a binary solution, the system may be assessed better in terms 
of three components, expressing the data on a triangular ternary equilibrium 
diagram (see section 4.6). 

3.9 Measurement of solubility 

Innumerable techniques, of almost infinite variety, have been proposed at one 
time or another for the measurement of the solubility of solids in liquids. No 
single method can be identified, however, as being generally applicable to all 
possible types of system. The choice of the most appropriate method for a given 
case has to be made in the light of the system properties, the availability of 
apparatus and analytical techniques, the skill and experience of the operators, 
the precision required, and so on. 

The accuracy required of a solubility measurement depends greatly on the 
use that is to be made of the information. Requirements vary enormously. In 
some cases, a simple assessment of whether a substance is highly, moderately or 
sparingly soluble in a given solvent, with some rough quantification, may be 
quite sufficient. In others, very high precisions may be demanded. For most 
work, however, a precision of < 1 % should be aimed for, and usually this is not 
too difficult to attain. 

Extensive reviews of the literature on the subject of experimental solubility 
determination have been made by Void and Void (1949) and Zimmerman 
(1952). Purdon and Slater (1946) give an excellent account of the determination 
of solubility in aqueous salt systems. The monographs of Blasdale (1927) and 
Teeple (1929) give comprehensive accounts of the problems encountered in 
measuring equilibria in complex multicomponent aqueous salt systems. 

Temperature control 

Constant temperature control is essential during all the experimental pro- 
cedures for solubility determination, not only during equilibration, but also 
during the sampling of saturated solution for analysis. The allowable limits of 
temperature variation depend on the system under investigation and the 
required precision of the solubility measurement. Much greater care has to be 
taken when the solubility changes appreciably with a change in temperature. In 
the determination of the solubility at 25 °C of, say, sodium chloride in water 
(Figure 3.1a), a variation of ±0.1 °C in the experimental temperature would 
allow for a potential precision of less than 0.01 per cent in the solubility, but the 
same temperature variation would allow for more than 1 per cent in the case of 
sodium sulphate (Figure 3.1b). For most general purposes, a thermostat preci- 
sion of better than ±0.1 °C is normally adequate. 

It is essential that any thermometers, thermocouples, thermistors, etc. used in 
the thermostat bath and equilibrium cell are accurately calibrated with refer- 
ence to a standard thermometer. This point cannot be emphasized too strongly. 

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Solutions and solubility 


Agitation of solutions 

Agitation is generally necessary to bring liquid and solid phases into intimate 
contact and facilitate equilibration. Agitation with a stirrer in an open vessel 
is not normally recommended, on account of the potential loss of solvent by 
evaporation, but sealed-agitated vessels are commonly used. Agitation in 
tightly stoppered vessels, that are rocked, rotated or shaken whilst immersed 
in a thermostat bath, is also quite a popular method, particularly when many 
samples have to be tested at the same time. 


Once equilibrium has been attained, i.e. when the originally over- and under- 
saturated solutions are of equal composition, the mixture is allowed to stand 
for an hour or more, at the relevant constant temperature, to enable any finely 
dispersed solid particles to settle. The withdrawal of a sample of clear super- 
natant liquid for analysis can be effected in a number of ways, depending on the 
characteristics of the system. For example, a suitably warmed pipette, with the 
tip protected by a piece of cotton wool, glass wool or similar substance, is often 
quite adequate. The pipette may be warmed to the appropriate temperature by 
leaving it standing in a stoppered tube immersed in the thermostat bath. 
Alternatively, a variety of sintered glass filters can be utilized (Figure 3.5) 
(Nyvlt, 1977). In all cases, several portions of solution should first be with- 
drawn and discarded to satisfy any possible capacity of the filter to adsorb 
solute from the saturated solution (Void and Void, 1949). The sample of 
saturated solution may then be analysed by any convenient technique (section 

It is important to note that a weighed quantity, not a measured volume, of 
solution should be taken for analysis. Weighing should be carried out to 
±0.0005 g if possible, depending on the required precision. 

l7^\^-, =< 






Figure 3.5. Some sintered glass filters for separating an equilibrium solid phase from 
a saturated solution. (After Nyvlt, 1977) 

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114 Crystallization 

Achievement of equilibrium 

The achievement of equilibrium presents one of the major experimental diffi- 
culties in solubility determination. Prolonged agitated contact is required 
between excess solid solute and solution at a constant temperature, usually 
for several hours. In some cases, however, contact for days or even weeks may 
be necessary. Viscous solutions and systems at relatively low temperatures 
often require long contact times and so do substances of low solubility. 

A check should always be made, if possible, on the accuracy of a solubility 
determination at a given temperature, by approaching equilibrium from both 
the under-saturated and over-saturated states. In the first method a quantity of 
solid, in excess of the amount required to saturate the solvent at the given 
temperature, is added to the solvent and the two are agitated until apparent 
equilibrium is reached, i.e. when the solution attains a constant composition. In 
the second method the same quantities of solute and solvent are mixed, but the 
system is then heated for about 20 minutes above the required temperature, if 
the solubility increases with temperature, so that most but not all of the solid is 
dissolved. The solution is then cooled and agitated for a long period at the 
given temperature while the excess solid is deposited and an apparent equi- 
librium is reached. If the two solubility determinations agree, it can be reas- 
onably assumed that the result represents the true equilibrium saturation 
concentration at the given temperature. If they do not, more time has to be 
allowed. This important point always has to be borne in mind when solubility 
measurements have to be made in solutions containing impurities (section 3.8). 

Unless solubility data for specific industrial substances are required, both the 
solute and solvent should be of the highest purity possible. The solute particles 
should be reasonably small to facilitate rapid dissolution, but not too small that 
the excess particles will not settle readily in the saturated solution. Settling is 
generally desirable to allow solid and liquid phase samples to be taken, after 
equilibration, for separate analysis. In practice, a close-sieved crystal fraction in 
the 100-300 um size range is generally suitable for most purposes. 

3.9.1 Solution and solid phase analysis 

Tremendous advances have been made in the past few decades in both the 
range and sensitivity of the analytical methods now available. For the purpose 
of solubility measurement, solution compositions can be measured by any 
convenient analytical technique, among which may be listed: liquid chromato- 
graphy (HPLC), spectroscopy (UV, IR, NMR and mass), differential scanning 
calorimetry (DSC), differential thermal analysis (DTA), thermogravimetric 
analysis (TGA), refractometry, polarimetry, and most recently capillary electro- 
phoresis (Altria, 2000). 

For the identification of crystalline polymorphs, IR spectroscopy and X-ray 
diffraction are the most commonly used techniques, while a combination of 
DSC-TGA or DTA-TGA are useful for analysing solvates. 

Descriptions of these techniques will be found in most handbooks of chem- 
ical and physical analysis (e.g. Findlay, 1973; Matthews, 1985). Comprehensive 

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Solutions and solubility 1 1 5 

accounts of modern instrumental methods of analysis is given by Willard, 
Merritt, Dean and Settle (1988) and Ewing (1997). 

When the dissolved substance is stable to heat, the mass present in a known 
mass of solution may often be determined by gentle evaporation to dryness, 
followed by heating to constant mass in an air oven at around 100 °C. Great 
care must be taken in this procedure to avoid solids loss by spattering. Another 
problem is that of solution creeping up the sides of the evaporating dish and 
over the edge. Gentle evaporation on a water bath may be used with aqueous 
solutions, covering the evaporating dish with a large funnel to prevent dust 

For substances that are not stable on heating to dryness, e.g. hydrates, their 
concentration in solution may be determined in some cases by chemical analysis 
and in others by measuring some concentration-dependent physical property. 
The latter is often convenient when a large number of determinations have to 
be made. Solution density and refractive index are probably the two properties 
most commonly measured for this purpose and the many well-established 
techniques available are fully described in standard textbooks of practical 
physical chemistry. 

In all cases, however, it is necessary to prepare a calibration chart. The first 
step, therefore, is to make up a series of solutions of known strength and 
measure the physical property in question for these solutions. This may be 
done for a range of temperatures, so that a series of calibration curves can be 
constructed. The composition of an unknown solution may then be determined 
by measuring the property in question at a given temperature, usually a few 
degrees above the saturation temperature to avoid the possibility of crystal- 
lization. Alternatively, a weighed quantity of the unknown solution is diluted 
with a known mass of solvent before measurement. 

It is always advisable, incidentally, to measure the density of a saturated 
solution at the same time that the equilibrium saturation concentration is being 
measured, for the simple practical reason that density is the mass-volume 
conversion factor, and this quantity is frequently required in process calcula- 
tions. Sohnel and Novotny (1985) have published an extensive compilation of 
concentration-density data for aqueous solutions of inorganic salts. 

The final step that has to be taken in a solubility determination, in order to 
complete the information, is to determine the composition of the solid phase 
that was in equilibrium with the solution at the given temperature, remember- 
ing that the stable phase can change appreciably over quite a short range of 
temperature, especially in hydrated systems. For example, in the determination 
of the solubility of sodium carbonate in water over the temperature range 
10-50 °C, it would be found that the stable solid phase is Na 2 C0 3 • 10H 2 O up 
to 32.0 °C, Na 2 C0 3 • 7H 2 between 32.0 and 35.4 °C and Na 2 C0 3 • H 2 above 

A sample of the equilibrium solid phase may be separated from its saturated 
solution by means of sintered glass filters like those depicted in Figure 3.5. It is 
necessary to ensure that the separation is made at the appropriate temperature, 
e.g. by carrying out the operation with the filter immersed in the thermostat 
bath itself. No matter how efficiently the filtration is made, however, the 

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116 Crystallization 

recovered solid particles will always be wet with adhering mother liquor and 
this must be accounted for in any subsequent analysis. First, it is essential to 
separate the wet solid sample as quickly as possible and transfer it immediately 
to a weighing bottle which can be closed to minimize loss of solvent. 

For simple binary systems, i.e. one solute and one solvent, correction for 
adhering mother liquor may be made in the following manner. The damp solid 
is weighed (mass m\) and then dried to constant mass, mi, at an appropriate 
temperature, taking care to avoid decomposition, dehydration of hydrates, etc. 
The amount of solvent present in the mother liquor, s (e.g. grams of solvent per 
gram of solution) is determined from the saturated solution analysis described 
above. The mass of mother liquor, mj, originally retained in the damp solid 
sample may then be calculated from 

m-i = (mi — mi)\s (3.64) 

The actual composition of the equilibrium solid phase may then be calculated 
from a simple mass balance, using the respective compositions of the damp 
solid and the equilibrium mother liquor. 

For multi-component systems the composition of the equilibrium solid phase 
may be determined indirectly by the so-called 'wet residues' method first 
proposed by Schreinemakers (1893) in which the need for solid-liquid separa- 
tion by filtration, etc. is avoided. The experimental procedures, together with 
those of the alternative 'synthetic complex' method, are fully described in 
section 4.6.5. 

3.9.2 Measurement techniques 

The so-called 'synthesis' methods of solubility determination involve the pre- 
paration of a solvent-solute mixture of known composition, initially containing 
excess solute. The complete dissolution of the solid phase is then observed, 
either when the mixture is subjected to slow controlled heating (the 'polythermal' 
methods) or at constant temperature when small quantities of fresh solvent 
are sequentially added over a period of time (the 'isothermal' methods). The 
disappearance of the solid phase can be observed visually or monitored by 
recording some appropriate physical or physicochemical property of the system. 

Polythermal methods 

Solute and solvent are weighed into a small (50-100 mL) glass vessel in propor- 
tions corresponding approximately to the composition of a saturated solution 
in the middle of the proposed operating temperature range. The objective is to 
have some solid phase in excess at the lowest temperature used and all in 
solution at the highest. 

The closed vessel is fitted with a calibrated thermometer graduated in incre- 
ments of 0.1 °C and a suitable stirrer, and the contents are heated gently until 
all the crystals have been dissolved. The clear solution is first cooled until it is 
nucleated. Then, under controlled conditions, the temperature is increased 

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Solutions and solubility 


slowly (^0.2°C/min) until the last crystal dissolves. At this point the equilib- 
rium saturation temperature 9* has been achieved. Repeat runs will enable 0* 
for a given solution composition to be determined with a precision of ±0.1 °C 
for solutions with a moderate to high temperature coefficient of solubility 
(section 3.2). 

An apparatus in which the controlled heating and cooling sequences 
demanded in the above technique is depicted in Figure 5.9 and described in 
section 5.3. Determination of the instant at which all the crystals have finally 
dissolved in a solution is most commonly made by visual observation. In 
principle, however, the monitoring of any concentration-sensitive physical or 
physicochemical property (refractive index, conductivity, density, vapour pres- 
sure, particle size distribution, etc.) can offer alternative procedures. 

Nyvlt (1977), for example, has described how refractive index measurements 
may be used for this purpose, where the sequence of events is as drawn in Figure 
3.6a. As the solution containing suspended crystals is heated, the refractive 
index of the solution increases as the crystals dissolve. At point A the last 
crystal dissolves, at the equilibrium saturation temperature 0* . Further 
increases in temperature lead to a slow decrease in refractive index. The reverse 
curve traces the cooling sequence, with nucleation occurring at point B, when 
the refractive index suddenly falls to point C and subsequently follows the 
equilibrium saturation curve. The corresponding solution composition- 
temperature graph is also included in Figure 3.6b. The refractive index may 
be monitored with a dipping-type refractometer. 

The property of refractive index may be utilized in another way. A novel 
technique was devised by Dauncy and Still (1952) for the direct and rapid 
measurement of solution saturation temperatures. This method is based on 
an optical effect caused by the slight change in concentration, and therefore in 
refractive index, occurring in a layer of solution immediately in contact with a 
crystal that is either growing or dissolving. 

The saturation cell is a small Perspex container fitted with a stirrer (altern- 
atively, the solution may be passed continuously through the cell), a calibrated 



8 /"" 








/ 1 


/v c 





Figure 3.6. Refractive index (a) and composition (b) changes during solubility 
measurement by the polythermal method. {After Nyvlt, 1977) 

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118 Crystallization 

I ^1 ^J 






(unsaturated ) 

( saturated ) 





Figure 3.7. An optical method of solubility measurement 

thermometer and a holder for a medium-sized crystal. The cell is placed in a 
thermostatically-controlled water bath also made of Perspex. A beam of light 
from an optical slit is directed on to an edge of the crystal, and the appearance 
of the slit when viewed from behind the crystal will take the form of one of the 
three sketches shown in Figure 3.7 . The light is bent into an obtuse angle when 
the solution is unsaturated (a), and into an acute angle when it is supersaturated 
(c). As soon as it is determined that the solution near the crystal face is 
unsaturated or supersaturated, the temperature is raised or lowered until view 
(b) is obtained. The temperature at this point is the equilibrium saturation 
temperature. It was reported that, with ethylenediamine hydrogen tartrate 
and ammonium dihydrogen phosphate, points on the respective solubility 
curves could be plotted at the rate of 8-10 per hour. Wise and Nicholson 
(1955) adapted the method and applied it successfully in the determination of 
sucrose solubilities. 

Isothermal methods 

The disappearance of the solid phase in a solubility cell can be observed under 
isothermal conditions while adding small portions of fresh solvent to a solu- 
tion-suspension of known composition. Mullin and Sipek (1982) have 
described one use of this technique for solubility measurements in the three- 
phase system potash-alum-water-ethanol. 

The apparatus used for the solubility determinations (Figure 3.8) was a small 
glass vessel (^50 mL) fitted with a four-blade glass stirrer with a glycerol shaft 
seal. The cell was immersed in a thermostat water bath controlled to ±0.02 °C. 
Weighed quantities of potash alum and alcohol, together with predetermined 
amounts of water, were charged to the cell and agitated for ~ 1 h. At the end of 
this time, as predicted, only a small amount of crystalline material was left 
undissolved. Small quantities of water (starting with 1 mL and reducing) were 
then added to the mixture at hourly intervals until all traces of crystalline 
material (observed under a strong back-light) had disappeared. Towards 
the end-point, water was added dropwise. This method, when carefully 
performed, could reproducibly determine the solubility to a precision of at least 

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Solutions and solubility 119 

Figure 3.8. Solubility apparatus: (a) 50 mL glass cell; (b) four-bladed glass stirrer, 
(c) charging port; (d) glycerol seal for stirrer shaft. (After Mullin and Sipek, 1982) 

Measurement under pressure 

An equilibrium cell for use under pressure is described by Brosheer and Ander- 
son (1946). Although originally devised specifically for solubility measurements 
of monoammonium and diammonium phosphates in the system NH3 — H2O— 
H3PO4, it is clearly of more general applicability. For a detailed account of this 
method, reference should be made to the original paper. A simple apparatus 
for the measurement of solubility under pressure was earlier described by 
Gibson (1934) who also made an interesting analysis of the pressure effect 
and demonstrated the possibility of estimating the solubility of certain salts in 
water under pressures up to lOkbar using data obtained at lower pressures. 

Thermal and dilatometric methods 

A phase reaction is generally accompanied by significant enthalpy and volume 
changes. The detection and quantification of these effects form the basis of 
several useful methods for determining solubilities and phase equilibria. Some 
of these techniques are discussed in section 4.5. 

Sparingly soluble salts 

The solubility of sparingly soluble electrolytes in water, with the exception of 
the salts of weak acids or bases, may be determined from conductivity measure- 
ments on their saturated solutions. A variety of commercial instruments are 
now available for this purpose and experimental details may be found in 
handbooks of practical physical chemistry, e.g. Findlay, 1973; Matthews, 1985. 

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120 Crystallization 

If the equilibrium saturation concentration, c* , of the salt is expressed in 
molm~ , the molar conductivity, A, (Sm 2 mol _1 ) of the solution may be 
expressed (see equation 3.45) as 

A = k/c* (3.65) 

where k is the conductivity (Sm~ ). Even a saturated solution of sparingly 
soluble salt is still very dilute, so it may be assumed that A ~ Ao where Ao is the 
molar conductivity at infinite dilution. If, therefore, the value of Ao is known, 
or can be calculated since it is the sum of the ionic conductivities (equation 
3.46), the saturation concentration of salt may be calculated from 

c* = k/A (3.66) 

where k is the measured conductivity of the saturated solution. 

For example, the conductivity of a saturated solution of barium sulphate in 
water at 25 °C has been experimentally determined as 1.66 x 10 _4 Sm _ . If the 
conductivity of the water used in the determination (2.5 x 10~ 5 S m~ ) is deducted 
from the measured solution conductivity, the value attributable to BaSCU is 
k = 1 .41 x 10~ 4 S m _1 . From Table 3.2, the relevant molar ionic conductivities at 
25 °C at infinite dilution are A 1/2Ba 2 = 63.6 x 10~ 4 Sm 2 moF and A^so 2 = 
78.8 x 10" 4 Sm 2 mor' giving A = 143.4 x 1(T 4 Sm 2 mor 1 . From equation 

c* = 1.41 x l(T 4 /143.4x 1(T 4 

= 9.83 x l(T 3 molirr 3 

= 9.83 x lO^molL" 1 

The corresponding solubility product K c may be calculated from equation 3.49 
(since BaSC>4 is a 2-2 electrolyte), i.e. 

Kc = {c*f = 9.66 x 1CT 11 mol 2 Lr 2 

3.10 Prediction of solubility 

A measured value of solubility, even when roughly determined, generally gives 
more confidence than an estimated one. Accurate solubility measurements, 
however, demand laboratory facilities and experimental skills (section 3.9) 
and can be very time-consuming on account of the need to achieve equilibrium 
and the fact that large numbers of individual measurements may be necessary 
to cover adequately all the ranges of variables. There will always be a need, 
therefore, for methods of solubility prediction that can avoid these difficulties, 
but it has to be pointed out that in employing such methods some other more 
serious problems may well be incurred in return. A good number of solubility 
correlation and prediction methods are available ranging from simple tech- 
niques of interpolation and extrapolation to some quite complex procedures, 
based on thermodynamic reasoning, that have considerable computational 

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Solutions and solubility 121 

The success of any given method can vary enormously from system to 
system. Some can only be used for rough assessment while others can occa- 
sionally yield data of comparable precision to those attained by careful experi- 
mental measurement. Each system must be considered independently. 

For binary solutions (one solute, one solvent) data correlations of the type 
indicated by equations 3.4 and 3.9 are commonly used for predicting, by 
interpolation, values that were not otherwise measured. If the correlating 
equation is based on adequate data, interpolation can be carried out with 
reasonable confidence (e.g. see Figure 3.2). Extrapolation, on the other hand, 
is generally inadvisable and should never be attempted if there is any suspicion 
that a phase change is possible in the unknown region. 

Prediction methods using theoretical relationships based on the assumption 
of solution ideality can be very unreliable, as shown by the example in section 
3.6 which indicates that an assumption of ideality for the 'simple' case of 
naphthalene dissolved in an organic solvent can result in an error of up to 
200% in estimating the solubility. 

The number of solubility measurements necessary for the construction of a 
multicomponent phase diagram increases enormously as the number of com- 
ponents is increased. It is in this area, therefore, that the demand for prediction 
method most often lies. The methods range from the entirely empirical, 
generally based on geometrical concepts, to the semi-theoretical, i.e. partly 
based on thermodynamic descriptions. A comprehensive account of some of 
these methods, together with several detailed worked examples, is given by 
Nyvlt (1977). 

Several thermodynamic approaches have been made to the problem of 
solubility predictions in multicomponent aqueous salt solutions over the past 
30 years or so, with varying degrees of success. Most methods are based to 
some extent on modified Debye-Hiickel equations (section 3.6.1) and require 
the prediction of activity coefficients, enthalpies and entropies of solution, and 
specific heat capacities. Within the confines of this chapter, however, it is not 
possible to give more than a flavour of the relevant literature in this area. 

For example, Marshall and Slusher (1966) made a detailed evaluation of the 
solubility of calcium sulphate in aqueous sodium chloride solution, and sug- 
gested that variations in the ion solubility product could be described, for ionic 
strengths up to around 2 M at temperatures from to 100 °C, by adding another 
term in an extended Debye-Hiickel expression. Above 2 M and below 25 °C, 
however, further correction factors had to be applied, the abnormal behaviour 
being attributed to an increase in the complexity of the structure of water under 
these circumstances. Enthalpies and entropies of solution and specific heat 
capacity were also reported as functions of ionic strength and temperature. 

A thermodynamic model developed by Barba, Brandani and di Giacomo 
(1982) described the solubility of calcium sulphate in saline water. A system of 
equations based on Debye-Hiickel and other models was used to describe 
isothermal activity coefficients of partially or completely dissociated electro- 
lytes. Using binary parameters, good agreement was claimed between experi- 
mental and predicted values of calcium sulphate solubility in sea water and 
brackish brines including those with a magnesium content. 

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122 Crystallization 

The solubilities of the scale-forming salts barium and strontium sulphates in 
aqueous solutions of sodium chloride have been reviewed by Raju and Atkin- 
son (1988, 1989). Equations were proposed for the prediction of specific heat 
capacity, enthalpy and entropy of dissolution, etc., for all the species in the 
solubility equilibrium, and the major thermodynamic quantities and equilib- 
rium constraints expressed as a function of temperature. Activity coefficients 
were calculated for given temperatures and NaCl concentrations and a com- 
puter program was used to predict the solubility of BaSC>4 up to 300 °C and 
SrS0 4 upto 125 °C. 

A group contribution method called UNIFAC, an acronym which stands for 
the UNIQUAC Functional Group Activity Coefficient (UNIQUAC stands 
for the Universal Quasi-chemical Activity Coefficient), has been developed 
for estimating liquid-phase activity coefficients in non-electrolyte mixtures. 
The UNIFAC method is fully described by Fredenslund, Jones and Prausnitz 
(1975) and Skold-Jorgensen, Rasmussen and Fredenslund (1982). 

To estimate the solubility of an organic solid solute in a solvent it is only 
necessary to know its melting point, enthalpy of fusion and relevant activity 
coefficient. Gmehling, Anderson and Prausnitz (1978) have shown that this 
activity coefficient can be estimated by the UNIFAC group contribution 
method, and they report a number of cases where the solubilities of a variety 
of organic solids in single and mixed solvents are accurately predicted. Even 
eutectic temperatures and compositions may be estimated for some binary 

Gupta and Heidemann (1990) used a modified UNIFAC model to predict 
the effects of temperature and pH on the solubility of amino acids in water. 
They also made a similar approach to the modelling of the solubility of several 
antibiotic substances in mixed non-aqueous solvents. Macedo, Skovborg and 
Rasmussen (1990) used a modified UNIFAC model to calculate phase equilib- 
ria for aqueous solutions of strong electrolytes. 

As explained in section 3.6.1, many modifications have been proposed for the 
Debye-Hiickel relationship for estimating the mean ionic activity coefficient j± 
of an electrolyte in solution and the Davies equation (equation 3.35) was 
identified as one of the most reliable for concentrations up to about 0.2 molar. 
More complex modifications of the Debye-Hiickel equation (Robinson and 
Stokes, 1970) can greatly extend the range of 7± estimation, and the Bromley 
(1973) equation appears to be effective up to about 6 molar. The difficulty with 
all these extended equations, however, is the need for a large number of 
interacting parameters to be taken into account for which reliable data are 
not always available. 

A more simple, but purely empirical, approach to the estimation of j± was 
suggested by Meissner and Tester (1972) who claimed applicability up to 
saturation or 20 molar. They noted that for over a 100 electrolytes a plot of 
a 'reduced' activity coefficient versus the ionic strength, I, formed a family of 
non-intersecting curves. They proposed methods of interpolation and extra- 
polation working from the basis of at least one known value of j± for a concen- 
trated solution of the chosen electrolyte. A survey of the use of this method, and 
its subsequent development for computer-assisted calculations (Meissner and 

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Solutions and solubility 123 

Manning, 1983), has been made by Demopoulos, Kondos and Papangelakis 

Vega and Funk (1974) presented a thermodynamic correlation for solid- 
liquid equilibria in concentrated aqueous salt solutions and applied the correla- 
tion to the six-component system containing Na + , K + , Mg 2+ , NO^~, SO4 from 
to 50 °C. In their correlation they define an activity coefficient as if the given 
salt were a non-electrolyte, although this new quantity is easily related to the 
mean ionic activity coefficient j±. The derived parameters are claimed to enable 
correlation of equilibria for ternary and quaternary systems with errors in 
liquid phase composition of less than 2g salt per lOOg water. 

3.11 Solubility data sources 

The main primary sources of solid-liquid solubility data, i.e. those which report 
experimental measurements together with the full literature source references, 
are those of Stephen and Stephen (1963), Seidell (1958) and the continuing 
multivolume IUPAC Solubility Data Series (1980-91) which by the end of 1991 
had reached its 48th volume. The series covers gas-liquid, liquid-liquid and 
solid-liquid equilibria, but up to the present time fewer than one quarter of the 
published volumes are devoted to solid-liquid systems. In all these publica- 
tions, ternary as well as binary data are reported and solvents other than water 
are considered. Blasdale (1927) and Teeple (1929) give extensive data on 
equilibria in aqueous salt solutions relevant to natural brines and natural salt 
deposits, ranging from binary to quinary complex systems. The compilation by 
Wisniak and Herskowitz (1984) is an excellent literature source reference, but 
no actual data are recorded. 

Among the secondary sources of data available, i.e. summaries assembled 
from several sources, sometimes 'smoothed', include the compilations of Nyvlt 
(1977), and Broul, Nyvlt and Sohnel (1981) and Appendices A4 and A5 in this 

3.12 Supersolubility 

A saturated solution is in thermodynamic equilibrium with the solid phase, at 
a specified temperature. It is often easy, however, e.g. by cooling a hot con- 
centrated solution slowly without agitation, to prepare solutions containing 
more dissolved solid than that represented by equilibrium saturation. Such 
solutions are said to be supersaturated. 

The state of supersaturation is an essential requirement for all crystallization 
operations. Ostwald (1897) first introduced the terms 'labile' and 'metastable' 
supersaturation to classify supersaturated solutions in which spontaneous (prim- 
ary) nucleation (see section 5.1) would or would not occur, respectively. The 
work of Miers and Isaac (1906, 1907) on the relationship between supersatura- 
tion and spontaneous crystallization led to a diagrammatic representation of 
the metastable zone on a solubility-supersolubility diagram (Figure 3.9). The 

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124 Crystallization 

Figure 3.9. The solubility-super solubility diagram 

lower continuous solubility curve, determined by one of the appropriate tech- 
niques described in section 3.9, can be located with precision. The upper broken 
supersolubility curve, which represents temperatures and concentrations at 
which uncontrolled spontaneous crystallization occurs, is not as well defined 
as that of the solubility curve. Its position in the diagram is considerably 
affected by, amongst other things, the rate at which supersaturation is gener- 
ated, the intensity of agitation, the presence of trace impurities and the thermal 
history of the solution (sections 5.3 and 5.4). 

In spite of the fact that the supersolubility curve is ill-defined, there is no 
doubt that a region of metastability exists in the supersaturated region above 
the solubility curve. The diagram is therefore divided into three zones, one well- 
defined and the other two variable to some degree: 

1. The stable (unsaturated) zone, where crystallization is impossible. 

2. The metastable (supersaturated) zone, between the solubility and super- 
solubility curves, where spontaneous crystallization is improbable. How- 
ever, if a crystal seed were placed in such a metastable solution, growth 
would occur on it. 

3. The unstable or labile (supersaturated) zone, where spontaneous crystal- 
lization is probable, but not inevitable. 

If a solution represented by point A in Figure 3.9 is cooled without loss of 
solvent (line ABC), spontaneous crystallization cannot occur until conditions 
represented by point C are reached. At this point, crystallization may be 
spontaneous or it may be induced by seeding, agitation or mechanical shock. 
Further cooling to some point D may be necessary before crystallization can be 
induced, especially with very soluble substances such as sodium thiosulphate. 
Although the tendency to crystallize increases once the labile zone is pen- 
etrated, the solution may have become so highly viscous as to prevent crystal- 
lization and could even set to a glass. 

Supersaturation can also be achieved by removing some of the solvent from 
the solution by evaporation. Line AB'C' represents such an operation carried 
out at constant temperature. Penetration beyond the supersolubility curve into 
the labile zone rarely happens, as the surface from which evaporation takes 
place is usually supersaturated to a greater degree than the bulk of the solution. 

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Solutions and solubility 125 

Crystals which appear on this surface eventually fall into the solution and seed 
it, often before conditions represented by point C' are reached in the bulk of the 
solution. In practice, a combination of cooling and evaporation is employed, 
and such an operation is represented by the line AB"C" in Figure 3.9. 

Experimental techniques for determining the metastable zone width, the 
amount of undercooling that a solution will tolerate before nucleating, are 
described in section 5.3. The significance of the metastable zone and the inter- 
pretation of metastable zone width measurements are somewhat contentious 
subjects. Experimental values depend very strongly on the method of detection 
of the onset of nucleation, but it is still possible to extract kinetic information 
on the nucleation process as well as on the growth behaviour of very small 
crystals. These topics are discussed in some detail in section 5.3. 

3.12.1 Expressions of supersaturation 

The supersaturation, or supercooling, of a system may be expressed in a 
number of different ways, and considerable confusion can be caused if the 
basic units of concentration are not clearly defined. The temperature must also 
be specified. 

Among the most common expressions of supersaturation are the concentra- 
tion driving force, Ac, the supersaturation ratio, S, and a quantity sometimes 
referred to as the absolute or relative supersaturation, a, or percentage super- 
saturation, lOOer. These quantities are defined by 

Ac = c - c* (3.67) 


a = — =S-l (3.69) 


where c is the solution concentration, and c* is the equilibrium saturation at the 
given temperature. 

The term supercooling, defined by 

Ad = 9* - 9 (3.70) 

is occasionally used as an alternative to the supersaturation, Ac, the two 
quantities being related through the local slope of the solubility curve, 
dc*/d6, by 

Ac = (dc7d69A0 (3.71) 

Of the above three expressions for supersaturation (equations 3.67 to 3.69) 
only Ac is dimensional, unless the solution composition is expressed in mole 
fractions or mass fractions. The magnitudes of these quantities depend on the 
units used to express concentration, as the following examples show. 

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126 Crystallization 

Example 1 

Potassium sulphate (mol. mass = 174) at 20 °C. The equilibrium saturation 
c* = 109 g of K2S04/kg of water, which gives a solution density of 1080 kg m -3 . 
Let the concentration of a supersaturated solution c= 116g/kg, giving a 
solution density of 1090 kg m -3 at 20 °C. Then the following quantities may 
be calculated: 

Solution composition c c* Ac S a 

g/kg water 116 109 7.0 1.06 0.06 

g/kg solution 104 98.3 5.7 1.06 0.06 

g/L solution (= kg m" 3 ) 113.3 106.1 7.2 1.07 0.07 

mol/L solution (=kmolm- 3 ) 0.650 0.608 0.042 1.07 0.07 

mol fraction of K 2 S0 4 0.0119 0.0112 0.0007 1.06 0.06 

It is essential to quote the temperature when expressing the supersaturation 
of a system, since the equilibrium saturation concentration is temperature 
dependent. In the above case of potassium sulphate, for example, S = 1.06 
means a concentration driving force Ac = 7 g/kg of water at 20 °C and 13 g/kg 

The quantity that changes most in example 1 is Ac; neither S nor a is very 
greatly affected. However, with very soluble substances considerable changes 
can occur in all expressions of supersaturation depending on the concentration 
units used, as seen in example 2 where a varies from 0.08 to 0.20. 

Example 2 

Sucrose (mol. mass = 342) at 20 °C, c* = 2040 g/kg of water (solution dens- 
ity = 1330 kgm~ 3 ). Let c = 2450 g/kg of water (density = 1360 kgm~ 3 ). Thus: 

Solution composition 






g/kg water 






g/kg solution 






g/L solution (=kgm~ 3 ) 






mol/L solution (= kmol m~ 3 ) 






mole fraction of sucrose 






The situation becomes even more confused than that hinted at in examples 
1 and 2 if the substance crystallizes as a hydrate, because in these cases solution 
compositions can be expressed in terms of the hydrate or the anhydrate, 
thus further increasing the number of possible definitions of supersaturation. 

Interconversion of solution composition units, as discussed in section 3.3, 
is facilitated by the formulae listed in Table 3.1. Interconversion of solution 
supersaturation values, based on seven different solution composition units, is 
facilitated by the formulae listed in Table 3.3 (Mullin, 1973). It should be clearly 

Table 3.3. Conversion factors for super saturation units (Mullin, 1973) 





s 2 


s 4 


"1 - C\(R - 1)" 

Equivalent expressions 


l + q 

l + C, 






1 - RC* 3 
1 -i?q 

rS 2 





s 3 



v l - 

c 3 

rS 3 

5 4 

j? + qiR- i) 

tf+qri?- i) 

l + q 

l + q 



l + q 
1 + 

s 5 

5 5 

0* - q 
p- c 5 

s s 





s 6 

S 6 


c fi 



5 7 

- m a c? 

M A q 


5 7 

M A q 

M A q 

Ci = kg of anhydrous substance/kg of water 

Ci = kg of anhydrous substance/kg of solution 

C3 = kg of hydrate/kg of solution 

Cn = kg of hydrate/kg of 'free' water 

C5 = kg of anhydrous substance/m 3 of solution 

C(, = kg of hydrate/m 3 of solution 

C7 = kmol of anhydrous substance/m 3 of solution 

Ma = molar mass of anhydrous substance 

Mu = molar mass of hydrate 

Mw = molar mass of water 

R = M H /M A 

p = density of supersaturated solution (kg/m 3 ) 

p* = density of saturated solution (kg/m 3 ) 

r = p/p* 

C3 c _, C( C-j 


Note: S 3 

- S? and S b 

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128 Crystallization 

appreciated, however, that none of these different supersaturations coincides 
exactly with the true thermodynamic supersaturation. 

The fundamental driving force for crystallization is the difference between 
the chemical potential of the given substance in the transferring and transferred 
states, e.g. in solution (state 1) and in the crystal (state 2). This may be written, 
for the case of an unsolvated solute crystallizing from a binary solution, as 

A/j, = n\ - H2 (3.72) 

The chemical potential, ri, is defined in terms of the standard potential, /zo, and 
the activity, a, by 

fi = fio + RT In a (3.73) 

where R is the gas constant and T is the absolute temperature. 

The fundamental dimensionless driving force for crystallization may there- 
fore be expressed as 

^=\n(a/a*) = \nS (3.74) 

where a* is the activity of a saturated solution and S is the fundamental super- 
saturation, i.e. 

S = exp(Azi/Rr) (3.75) 

For electrolyte solutions it is more appropriate to use the mean ionic activity, 
a±, defined by 

a = a± (3.76) 

where v(=v + + i>_) is the number of moles of ions in 1 mole of solute (equation 
3.30). Therefore, 

Aii/RT=v\nS a (3.77) 


S a = a±/a* ± (3.78) 

Alternatively, the supersaturation may be expressed as 

° a = S a - 1 (3.79) 

and equation 3.77 as 

A^/Rr=^ln(l+o- fl ) (3.80) 

For low supersaturations (a a < 0.1) 

An/RT & va a (3.81) 

is a valid approximation. 

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Solutions and solubility 129 

However, for practical purposes, supersaturations are generally expressed 
directly in terms of solution concentrations, e.g. 

S c = —, S m = — and S x = — (3.82) 

c* m* x* 

where c = molarity (mol/litre of solution), m = molality (mol/kg of solvent) 
and x = mole fraction. The asterisks denote equilibrium saturation. 

The relationship between these concentration-based supersaturations and the 
fundamental (activity-based) supersaturation may be expressed through the 
relevant concentration-dependent activity coefficient ratio, A = 7/7*, i.e. 

o a = £> C A C = S m A m = o x A x (j.oj) 

where A c = 7J7*, A m = 7^/7^ and A x = j x /l* x - 

If the relevant activity coefficients can be evaluated, it is possible to establish 
how the different supersaturations differ from one another and, more import- 
antly, from the fundamental supersaturation, S a . The decisive factor is the 
activity coefficient ratio, A. The more it deviates from unity, the greater is the 
incurred inaccuracy. In general, when A m > 1, m-based concentration units are 
preferred, but when A m < 1, x- or obased units are better than w-based. The 
choice between x- and c-based units in this case again depends on the activity 
coefficient ratio: if A x > A c , the x-based units are preferred and vice versa. 

For example, the mean ionic activity coefficients (see section 3.6.2) for a 
saturated solution of KC1 in water at 25 °C (m* = 4.761 molKCl/kg water) 
(Robinson and Stokes, 1970) are 

j* ±c = 0.6938; f ±m = 0.5923; 7^ = 1.013 

and those corresponding to a solution at the same temperature of concentration 
m = 5.237 molKCl/kg water, i.e. of supersaturation S m = 1.1 (Mullin and 
Sohnel, 1977) are 

7 ±c = 0.7157; j ±m = 0.6019; 1±x = 1.030 
The respective activity coefficient ratios {A = 7/7*) are, therefore, 

A c = 1.032; A m =1.016; A x =1.017 
so from equation 3.83 

S c = 0.9695 fl ; S m = 0.984S a ; S x = 0.9835 fl 

which indicates that in this case the supersaturation expressed on a molar basis 
is the least reliable. 

In general, in the absence of any information on the activity coefficient ratio, 
preference should be given to supersaturations based on molal units because of 
their more practical utility compared with mole fractions and their temperature 
independence compared with molar units. In other words, a concentration scale 
based on mass of solvent is generally preferred to one based on volume of 

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130 Crystallization 

An extension of the above analysis to more complex cases (Sohnel and 
Mullin, 1978a) leads to the conclusion that the dimensionless driving force 
for crystallization of a hydrate should always be expressed in terms of the 
hydrate and not of the anhydrous salt, i.e. 

(A M /Rr) H = ^ln5 H ^ H (3.84) 

where the solution concentrations and activity coefficients and their ratio An 
both relate to the hydrate. The difference between the hydrate (H) and anhy- 
drous (A) quantities, (A^/RT) H and (Ari/RT) A , can be very considerable. 
When, for lack of information, the quantity A^ cannot be evaluated, the 

(A/x/Rr) H «i/lnS H (3.85) 

may be used. No general rules have yet been derived about the preference for c-, 
m- or x-based concentration units for the expression of the supersaturation Sh in 
these cases, although the subject is further discussed by Sohnel and Garside (1992). 

Sparingly soluble electrolytes 

Supersaturations in aqueous solutions of sparingly soluble electrolytes are best 
expressed in terms of the solubility product, e.g. 

S = (IAP/K a y lv (3.86) 

where IAP is the ion activity product of the lattice ions in solution, K a is the 
activity solubility product of the salt, i.e., the value of IAP at equilibrium as 
defined in section 3.6.4, and v is the number of ions in a formula unit of the salt. 

When applying equation 3.86 to express the level of supersaturation created 
before the onset of precipitation, it is important to recognize that the values of 
IAP and K a used should be those appropriate to the conditions existing in the 
actual mother liquor at the completion of the precipitation reaction, and not to 
those relating to equilibria between the pure precipitate and pure solvent (water) 
which is the basis on which solubility products are normally listed, as in Table A3. 

A simple example of the magnitude of the error that can be incurred using 
the incorrect solubility values is demonstrated by the precipitation of BaSC>4 
after mixing equal volumes of 1 molar aqueous solutions of BaC^ and H2SO4 
at 10 °C, thus producing an initial mixture containing 0.5molL~' BaSC>4 and 
ImolL -1 HC1. The equilibrium solubility of BaS04 at 10 °C in water is 
8.56 x 10 _6 molL _1 , but in aqueous 1 molar HC1 it is 2.36 x 10 _4 molL _1 . 
So the initial supersaturation (equation 3.68) of BaSC>4 with respect to water 
is (0.5/8.56) x 10~ 6 = 58400, whereas that expressed, more correctly, with 
respect to solution in 1 molar HC1 is (0.5/2.36) x 10~ 4 = 2120 (Sohnel and 
Garside, 1992). 

A further difficulty in establishing the correct supersaturation with some 
systems is the necessity to determine the extent of any ion association, complex 
formation and hydration in the supersaturated solution. For example, in sys- 
tem such as CaC03-H20, in addition to the presence of Ca 2+ and C0 3 2 ions, 
others such as HCO^ and CaHCO^ must also be taken into consideration. The 

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Solutions and solubility 131 

calcium phosphate systems show even greater complexity. Good examples of 
the iterative calculations necessary to identify the correct value of IAP to be 
employed in equation 3.86 are given by Nancollas and Gardner (1974) and 
Barone and Nancollas (1977). 

Mixed salt systems 

It is not easy to quantify precisely the supersaturation levels of given species 
generated in mixed salt systems or in solutions in which ion association occurs. 
Relationships such as equations 3.67-3.69 cannot be simply applied because 
of the difficulty of expressing the true reference condition of equilibrium satura- 
tion. It is first necessary to identify all the possible single species, ion pairs and 
solid-liquid phase equilibria that can occur in the system. The relevant thermo- 
dynamic association/dissociation constants (K values) must be known. The 
activity coefficients for the various ionic species must be calculated, e.g. by 
means of Debye-Hiickel type equations (section 3.6.2). Equilibrium concentra- 
tions of all the possible species present are then evaluated by iterative procedures. 
Examples of these complex computing procedures are given in several pub- 
lications, e.g., for calcium carbonate (Wiechers, Sturrock and Marias, 1975), 
calcium phosphate (Barone and Nancollas, 1977), calcium oxalate (Nancollas 
and Gardner, 1974) and magnesium hydroxide (Liu and Nancollas, 1973), in 
a variety of electrolyte solutions. 

3.12.2 Measurement of supersaturation 

If the concentration of a solution can be measured at a given temperature, and 
the corresponding equilibrium saturation concentration is known, then it is 
a simple matter to calculate the supersaturation (equations 3.67-3.69). Just as 
there are many methods of measuring concentration (section 3.9.2) so there are 
also many ways of measuring supersaturation, but not all of these are readily 
applicable to industrial crystallization practice. 

Solution concentration may be determined directly by analysis, or indirectly 
by measuring some property of the system that is a sensitive function of solute 
concentration. The properties most frequently chosen for this purpose are 
density and refractive index which can often be measured with high precision, 
especially if the actual measurement is made under carefully controlled condi- 
tions in the laboratory. 

For the operation of a crystallizer under laboratory or pilot plant conditions 
the demand is usually for an in situ method, preferable one capable of con- 
tinuous operation. In these circumstances problems may arise from the temper- 
ature dependence of the property being measured. Nevertheless, the above 
properties can be measured, more or less continuously, with sufficient accuracy 
for supersaturation determination. 

The supersaturation of a concentrated solution may be determined from a 
knowledge of its boiling point elevation. Holven (1942) applied the principle of 
Diihring's rule (the boiling point of a solution is a linear function of the boiling 
point of the pure solvent at the same pressure) to sucrose solutions over the 

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Absolute pressure, mbor 
100 200 300 400 600 

Duhring line 
for water 

S < I = unsaturated 
5=1 = saturated 
S=- I = supersaturated 

50 60 TO 80 

Boiling point of water, °C 


Figure 3.10. Diihring-type plot showing constant super saturation lines (range S = to 1.8) 
for aqueous solutions of sucrose. (After Holven, 1942) 

range of pressures normally encountered in sugar boiling practice (Figure 3.10) 
and developed an automatic method for recording and controlling the degree of 
supersaturation in sugar crystallizers. 

3.13 Solution structure 

Water is a unique liquid. It is also the most abundant compound on earth 
(~10 21 kg in the oceans with perhaps a similar quantity bound up as water of 
crystallization in rocks and minerals) and it is an essential constituent of all living 
organisms. Its unusual properties, such as a high boiling point compared with its 
related hydrides, a high thermal conductivity, dielectric constant and surface 
tension, a low enthalpy of fusion, the phenomenon of maximum density (at 
4 °C), etc., are usually explained by assuming that liquid water has a structure. 

It is not possible at the present time to decide conclusively between the 
various structural models that have been proposed, but there is no doubt that 
liquid water does retain a loose local structure for short periods maintained by 
hydrogen bonds disposed tetrahedrally around each oxygen atom. Hydrogen 
bonded clusters readily form, but their lifetime is short (probably ^10~ n s); 
and the name 'flickering clusters' is particularly apt. 

The presence of a solute in water alters the liquid properties profoundly. In 
aqueous solutions of electrolytes, for example, the coulombic forces exerted 
by the ions lead to a local disruption of the hydrogen bonded structure. Each 
ion is surrounded by dipole orientated water molecules firmly bonded in what 
is known as the 'primary hydration sphere'. For monatomic and monovalent 
ions, four molecules of water most probably exist in the firmly fixed layer. For 

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Solutions and solubility 133 

polyvalent ions such as Cr 3+ , Fe 3+ and Al 3+ , six is a common number. The 
hydrated proton H30 + most probably exists as H30(H20)^. 

The electrostatic effects of an ion, however, can extend far beyond the 
primary hydration sphere. This accounts for the very large so-called hydration 
numbers that have been reported for some ions (up to 700 for Na + , for 
example). There is clearly a much larger region around the ion which contains 
loosely bound, but probably non-orientated, water. This assembly constitutes 
the 'secondary hydration sphere'. 

Some interesting comments have been made by Wojciechowski (1981) on 
evidence for structure in saturated aqueous solutions. An analysis of the 
solubilities of a number of inorganic salts in water, together with views on the 
structure of water, suggested a statistical concentration of phase transitions 
near certain temperatures, e.g. 30, 45 and 60 °C, giving a possible explanation 
for changes in the number of waters of crystallization in hydrates crystallizing 
around these temperatures. 

Detailed accounts of current theories of liquid structure are given by Samoi- 
lov (1965), Franks and Ives (1966), Franks (1972-82) and in Faraday Discus- 
sions (1967, 1978). 

Solute clustering 

The structure of a supersaturated solution is probably more complex than that 
of an unsaturated or saturated solution. As reported by Khamskii (1969) a 
number of attempts have been made to find the distinguishing features of super- 
saturated solutions by investigating the dependences of various physical proper- 
ties on concentration. In most cases, however, no evidence of discontinuity of 
the property-concentration curves at the equilibrium saturation point has been 
found, although an observation that light transmittancy could decrease sharply 
in the supersaturated region was regarded as evidence for solute clustering. 

Table 3.4. Concentration gradients developed in quiescent aqueous citric acid solutions 
kept under isothermal conditions (Mullin and Led, 1969a) 



after time t 

on all three 





































































c = solution concentration (g of citric acid monohydrate/g of 'free' water). 

c* = equilibrium saturation concentration (g of citric acid monohydrate/g of 'free' water). 

S = supersaturation ratio = c/c*. 

f = vertical distance between the sample points = 20 cm. 

f = unsaturated solution (one of many similar runs). 

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134 Crystallization 

Mullin and Leci (1969a) reported that supersaturated aqueous solutions 
of citric acid, kept quiescent at constant temperature, develop concentration 
gradients with the highest concentrations in the lower regions {Table 3.4). This 
unusual behaviour was taken as a strong indication of the existence of mole- 
cular clusters in such solutions, which is perhaps not unexpected in this case 
because citric molecules by virtue of their — OH and — COOH groups are 
capable of extensive hydrogen bonding between themselves and with the sol- 
vent water molecules. Using a similar technique, Allen et al. (1972) observed 
concentration gradients in supersaturated solutions of sucrose and developed a 
'settling' equation based on thermodynamic considerations. 

Larson and Garside (1986) reported on work with several supersaturated 
aqueous solutions, including citric acid, urea and sodium nitrate. Treating the 
cluster concentration as the solute concentration in excess of saturation, they 
developed a relationship which was in some respects similar to that used by Allen 
et al. (1972) from which cluster size estimates were made between 4 and 10 nm, 
containing up to 7000 molecules. In analysis based on non-equilibrium thermo- 
dynamics, Veverka, Sohnel, Bennema and Garside (1991) have suggested that 
concentration gradients could be expected to develop in quiescent columns of 
supersaturated solutions whether or not clustering occurs. The phenomenon of 
solute clustering, however, is still fully compatible with their proposed theory. 

Several attempts have been made to use Raman spectroscopy to estimate the 
degree of ionic and/or molecular association in supersaturated aqueous salt 
solutions. From the Raman spectra of ammonium dihydrogen phosphate 
solutions, Cerrata and Berglund (1987) concluded that whilst low-order (mono- 
mers and dimers) and high-order species were present, none of the clusters 
exhibited crystalline properties. A similar conclusion was reached by Rush, 
Schrader and Larson (1989) in a study on supersaturated solutions of sodium 
nitrate. In fact, the concentrated solution spectra were found to be very similar 
to those of sodium nitrate melts. 

The diffusivity of electrolytes and non-electrolytes in aqueous solution 
increases steadily with increasing concentration up to near the equilibrium 
saturation point, as shown by the data for NH4CI and KC1 in Figure 2.7. 
However, Myerson and his co-workers have demonstrated that above the 
saturation limit the diffusivity declines very rapidly. This is to be expected since 
supersaturated solutions are metastable and the diffusivity falls to zero at the 
spinodal, i.e., at the limit of the metastable zone (section 5.1.1). Diffusivity was 
also shown to decrease with solution age. All these observations are compatible 
with cluster theory, and analyses by Lo and Myerson (1990) and Ginde and 
Myerson (1992) suggest that clusters in supersaturated aqueous solutions of 
glycine are mainly in the form of dimers and trimers, although a few up to 100 
molecules can exist. Mohan, Kaytancioglu and Myerson (2000) also found 
trimer clusters in highly supersaturated solutions of ammonium sulphate and 
observed that the true metastable zone was much wider than had previously 
been thought, suggesting that virtually all bulk experiments involve heteroge- 
neous rather than homogeneous nucleation. Further comments on clusters and 
their role as nucleation precursors are made in section 5.1. 

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4 Phase equilibria 

4.1 The phase rule 

The amount of information which the simple solubility diagram can yield is 
strictly limited. For a more complete picture of the behaviour of a given system 
over a wide range of temperature, pressure and concentration, a phase diagram 
must be employed. This type of diagram represents graphically, in two or three 
dimensions, the equilibria between the various phases of a system. The Phase 
Rule, developed by J. Willard Gibbs in 1876, relates the number of compon- 
ents, C, phases, P, and degrees of freedom, F, of a system by means of the 

P+F= C+2 

These three terms are defined as follows. 

The number of components of a system is the minimum number of chemical 
compounds required to express the composition of any phase. In the system 
water-copper sulphate, for instance, five different chemical compounds can 
exist, viz. CuS0 4 • 5H 2 0, CuS0 4 • 3H 2 0, CuS0 4 • H 2 0, CuS0 4 and H 2 0; but 
for the purpose of applying the Phase Rule there are considered to be only two 
components, CuS0 4 and H 2 0, because the composition of each phase can be 
expressed by the equation 

CuS0 4 + xH 2 ^ CuS0 4 • xH 2 

Again, in the system represented by the equation 

CaC0 3 ^ CaO + C0 2 

three different chemical compounds can exist, but there are only two compon- 
ents because the composition of any phase can be expressed in terms of the 
compounds CaO and C0 2 . 

A phase is a homogeneous part of a system. Thus any heterogeneous system 
comprises two or more phases. Any mixture of gases or vapours is a one-phase 
system. Mixtures of two or more completely miscible liquids or solids are also 
one-phase systems, but mixtures of two partially miscible liquids or a hetero- 
geneous mixture of two solids are two-phase systems, and so on. 

The three variables that can be considered in a system are temperature, 
pressure and concentration. The number of these variables that may be changed 
in magnitude without changing the number of phases present is called the 
number of degrees of freedom. In the equilibrium system water-ice-water vapour 
C = 1, P = 3, and from the Phase Rule, F = 0. Therefore in this system there are 
no degrees of freedom: no alteration may be made in either temperature or 
pressure (concentration is obviously not a variable in a one-component system) 
without a change in the number of phases. Such a system is called 'invariant'. 

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For the system water-water vapour C = 1 , P = 2 and F = 1 : thus only one 
variable, pressure or temperature, may be altered independently without chan- 
ging the number of phases. Such a system is called 'univariant'. The one-phase 
water vapour system has two degrees of freedom; thus both temperature and 
pressure may be altered independently without changing the number of phases. 
Such a system is called 'bivariant'. 

Summarizing, it may be said that the physical nature of a system can be 
expressed in terms of phases, and that the number of phases can be changed by 
altering one or more of three variables: temperature, pressure or concentration. 
The chemical nature of a system can be expressed in terms of components, and 
the number of components is fixed for any given system. 

Comprehensive accounts of the phase rule and its applications have been 
given by Bowden (1950), Findlay and Campbell (1951), Ricci (1966), Haase and 
Schonert (1969) and Nyvlt (1979). 

4.2 One-component systems 

The two variables that can affect the phase equilibria in a one-component, or 
unary, system are temperature and pressure. The phase diagram for such a 
system is therefore a temperature-pressure equilibrium diagram. 

Figure 4.1 illustrates the equilibria between the vapour, liquid and solid 
phases of water. Curve AB, often referred to as the sublimation curve, traces 
the effect of temperature on the vapour pressure of ice. Curve BC is the vapour 
pressure curve for liquid water, and line BD indicates the effect of pressure on 
the melting point of ice, i.e. the freezing point of water. Water is an unusual 
substance in that it expands on freezing, indicated by the slope of line BD 
towards the left of the diagram, i.e. pressure decreases the melting point. The 

Figure 4.1. Phase diagram for water (not to scale) 

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Phase equilibria 


vast majority of other substances behave in the opposite manner (see Figure 
2.10). The three curves meet at the triple point B (0.01 °C, 6.1 mbar (610 Pa)) 
where ice, liquid water and water vapour can coexist in equilibrium. At the 
critical point C (374 °C, 220 bar (22 MPa)) liquid and vapour phases become 
indistinguishable. Above the critical point water is referred to as a supercritical 

The solvent properties of supercritical fluids are particularly interesting. 
Liquid water, for example, has a dielectric constant of around 80, whereas 
the value for supercritical water is around 2. At this low value it no longer acts 
as a polar solvent and many organic compounds can be dissolved in and 
crystallized from it. The potential exploitation of supercritical fluids, especially 
H2O and CO2, in crystallization processes is discussed in section 7.1.4. 

4.2.1 Polymorphs 

Figure 4.2 illustrates the case of sulphur, a system that exhibits two crystalline 
polymorphs. The area above the curve ABEF is the region in which ortho- 
rhombic sulphur is the stable solid form. The areas bounded by curves ABCD 
and FECD indicate the existence of vapour and liquid sulphur, respectively. 
The 'triangular' area BEC represents the region in which monoclinic sulphur is 
the stable solid form. Curves AB and BC are the vapour pressure curves for 
orthorhombic and monoclinic sulphur, respectively, and these curves intersect 
at the transition point B. 

Curve BE indicates the effect of pressure on the transition temperature for 
orthorhombic S ^ monoclinic S. Point B, therefore, is a triple point represent- 
ing the temperature and pressure (95.5 °C and 0.51 Nm~ 2 at which orthorhom- 
bic sulphur and sulphur vapour can coexist in stable equilibrium. Curve EF 
indicates the effect of pressure on the melting point of orthorhombic sulphur; 






M ^ 

Mire — 

A' J^>^ 

A Vapour 


Figure 4.2. Phase diagram for sulphur {not to scale) 

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138 Crystallization 

point E is a triple point representing the temperature and pressure (151 °C and 
1.31 x 10 8 Nm~ 2 ) at which orthorhombic and monoclinic sulphur and liquid 
sulphur are in stable equilibrium. Curve CD is the vapour pressure curve for 
liquid sulphur, and curve CE indicates the effect of pressure on the melting 
point of monoclinic sulphur. Point C, therefore, is another triple point (115°C 
and 2.4 Nm~ 2 ) representing the equilibrium between monoclinic and liquid 
sulphur and sulphur vapour. 

The broken lines in Figure 4.2 represent metastable conditions. If ortho- 
rhombic sulphur is heated rapidly beyond 95.5 °C, the change to the monoclinic 
form does not occur until a certain time has elapsed; curve BB', a continuation 
of curve AB, is the vapour pressure curve for metastable orthorhombic sulphur 
above the transition point. Similarly, if monoclinic sulphur is cooled rapidly 
below 95.5 °C, the change to the orthorhombic form does not take place 
immediately, and curve BA' is the vapour pressure curve for metastable mono- 
clinic sulphur below the transition point. Likewise, curve CB' is the vapour 
pressure curve for metastable liquid sulphur below the 115°C transition point, 
and curve B'E the melting point curve for metastable orthorhombic sulphur. 
Point B', therefore, is a fourth triple point (110°C and 1.7 Nm~ 2 ) of the system. 

Only three of the four possible phases orthorhombic (solid), monoclinic 
(solid), liquid and vapour can coexist in stable equilibrium at any one time, 
and then only at one of the three 'stable' triple points. 


The transformation from one polymorph to another can be reversible or 
irreversible; in the former case the two crystalline forms are said to be enantio- 
tropic; in the latter, monotropic. These phenomena, already described in sec- 
tion 1.8, can be demonstrated with reference to the pressure-temperature phase 

Figure 4.3a shows the phase reactions exhibited by two enantiotropic solids, 
a and j3. AB is the vapour pressure curve for the a form, BC that for the (3 
form, and CD that for the liquid. Point B, where the vapour pressure curves of 
the two solids intersect, is the transition point; the two forms can coexist in 
equilibrium under these conditions of temperature and pressure. Point C is 
a triple point at which vapour, liquid and (3 solid can coexist. This point can be 
considered to be the melting point of the (3 form. 

If the a solid is heated slowly, it changes into the (3 solid and finally melts. 
The vapour pressure curve ABC is followed. Conversely, if the liquid is cooled 
slowly, the (3 form crystallizes out first and then changes into the a form. Rapid 
heating or cooling, however, can result in a different behaviour. The vapour 
pressure of the a form can increase along curve BB' , a continuation of AB, the 
a form now being metastable. Similarly, the liquid vapour pressure can fall 
along curve CB', a continuation of DC, the liquid being metastable. Point B', 
therefore, is a metastable triple point at which the liquid, vapour and a solid 
can coexist in metastable equilibrium. 

The type of behaviour described above is well illustrated by the case of 
sulphur (Figure 4.2), where the orthorhombic and monoclinic forms are 

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Phase equilibria 139 


,/ n 













Figure 4.3. Pressure-temperature diagrams for dimorphous substances: (a) enantiotropy; 
(b) monotropy 

enantiotropic; the transition point occurs at a lower temperature than does the 
triple point. 

Figure 4.3b shows the pressure-temperature curves for a monotropic sub- 
stance. AB and BC are the vapour pressure curves for the a solid and liquid, 
respectively, and A'B' is that for the (3 solid. In this case the vapour pressure 
curves of the a and (3 forms do not intersect, so there is no transition point 
within this range of temperature and pressure. The solid form with the higher 
vapour pressure at any given temperature (J3 in this case) is the metastable 
form. Curves BB' and BB" are the vapour pressure curves for the liquid and 
metastable a solid, so B' is a metastable triple point. If this system did exhibit a 
true transition point, it would lie at point B"; but as this represents a tempera- 
ture higher than the melting point of the solid, it cannot exist. 

A typical case of monotropy is the change from white to red phosphorus. 
Benzophenone is another example of a monotropic substance: the stable melt- 
ing point is 49 °C, whereas the metastable form melts at 29 °C. 

The kinetics of polymorphic transformations in melts and solutions are 
discussed in section 6.5. 

4.3 Two-component systems 

The three variables that can affect the phase equilibria of a binary system are 
temperature, pressure and concentration. The behaviour of such a system 
should, therefore, be represented by a space model with three mutually perpen- 
dicular axes of pressure, temperature and concentration. Alternatively, three 
diagrams with pressure-temperature, pressure-concentration and temperature- 
concentration axes, respectively, can be employed. However, in most crystal- 
lization processes the main interest lies in the liquid and solid phases of 
a system; a knowledge of the behaviour of the vapour phase is only required 

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when considering sublimation processes. Because pressure has little effect on 
the equilibria between liquids and solids, the phase changes can be represented 
on a temperature-concentration diagram; the pressure, usually atmospheric, is 
ignored. Such a system is said to be 'condensed', and a 'reduced' phase rule can 
be formulated excluding the pressure variable: 

P+F' = C+ 1 

where F' is the number of degrees of freedom, not including pressure. 

Four different types of two-component system will now be considered. 
Detailed attention is paid to the first type solely to illustrate the information 
that can be deduced from a phase diagram. It will be noted that the concentra- 
tion of a solution on a phase diagram is normally given as a mass fraction or 
mass percentage and not as 'mass of solute per unit mass of solvent', as 
recommended for the solubility diagram (section 3.3). Mole fractions and mole 
percentages are also suitable concentration units for use in phase diagrams. 

4.3.1 Simple eutectic 

A typical example of a system in which the components do not combine to form 
a chemical compound is shown in Figure 4.4. Curves AB and BC represent the 
temperatures at which homogeneous liquid solutions of naphthalene in benzene 
begin to freeze or to crystallize. The curves also represent, therefore, the tem- 
peratures above which mixtures of these two components are completely liquid. 
The name 'liquidus' is generally given to this type of curve. In aqueous systems 
of this type one liquidus is the freezing point curve, the other the normal 
solubility curve. Line DBF represents the temperature at which solid mixtures 



O \ B 
" Solution + 
solid CgHg 

Yy Solution 

solid" C |0 H 3 



20 40 60 80 

Moss per cent naphthalene 


Figure 4.4. Phase diagram for the simple eutectic system naphthalene-benzene 

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Phase equilibria 141 

of benzene and naphthalene begin to melt, or the temperature below which 
mixtures of these two components are completely solid. The name 'solidus' is 
generally given to this type of line. The melting or freezing points of pure 
benzene and naphthalene are given by points A (5.5 °C) and C (80.2 °C), respect- 
ively. The upper area enclosed by the liquidus, ABC, represents the homo- 
geneous liquid phase, i.e. a solution of naphthalene in benzene; that enclosed 
by the solidus, DBE, indicates solid mixtures of benzene and naphthalene. The 
small and large 'triangular' areas ABD and BCE represent mixtures of solid 
benzene and solid naphthalene, respectively, and benzene-naphthalene solution. 

If a solution represented by point x is cooled, pure solid benzene is deposited 
when the temperature of the solution reaches point X on curve AB. As solid 
benzene separates out, the solution becomes more concentrated in naphthalene 
and the equilibrium temperature of the system falls, following curve AB. If 
a solution represented by point y is cooled, pure solid naphthalene is deposited 
when the temperature reaches point Y on the solubility curve; the solution 
becomes more concentrated in benzene and the equilibrium temperature fol- 
lows curve CB. Point B, common to both curves, is the eutectic point (— 3.5°C 
and 0.189 mass fraction of naphthalene), and this is the lowest freezing point in 
the whole system. At this point a completely solidified mixture of benzene and 
naphthalene of fixed composition is formed: it is important to note that the 
eutectic is a physical mixture, not a chemical compound. Below the eutectic 
temperature all mixtures are solid. 

If the solution y is cooled below the temperature represented by point Y on 
curve BC to some temperature represented by point z, the composition of the 
system as a whole remains unchanged. The physical state of the system has been 
altered, however; it now consists of a solution of benzene and naphthalene 
containing solid naphthalene. The composition of the solution, or mother 
liquor, is given by point z on the solubility curve, and the proportions of solid 
naphthalene and solution are given, by the so-called 'mixture rule', by the ratio 
of the lengths zZ and zZ', i.e. 

mass of solid CioH 8 zZ 

mass of solution zZ' 

A process involving both cooling and evaporation can be analysed in two steps. 
The first is as described above, i.e. the location of points z, Z and Z'; this 
represents the cooling operation. If benzene is evaporated from the system, z no 
longer represents the composition; thus the new composition point z' (not 
shown in the diagram) is located along line ZZ' between points z and Z. Then 
the ratio z'Z\z'Z' gives the proportions of solid and solution. 

The systems KCI-H2O and (NH/t^SOzt-t^O are good examples of aqueous 
salt solutions that exhibit simple eutectic formation. In aqueous systems the 
eutectic mixture is sometimes referred to as a cryohydrate, and the eutectic 
point a 'cryohydric point'. 

It should be understood that the term 'pure' when commonly used, as in this 
chapter, does not mean absolute 100% purity. In industrial crystallization 
practice this is neither necessary nor indeed achievable, and for many bulk- 
produced chemicals a purity of >95% is often accepted as justifying the 

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designation 'pure'. In any case, a single crystallization step cannot produce 
100% pure crystals for a variety of reasons, e.g., they can be contaminated with 
residual solvent or other impurities that have not been removed by washing, or 
have been incorporated into the crystal interstitially or as liquid inclusions, and 
so on. Furthermore, contamination commonly results from the existence of 
terminal solid solutions, which inevitably accompany both eutectic and chemical 
compound systems, as described in section 7.2. 

4.3.2 Compound formation 

The solute and solvent of a binary system may, and frequently do, combine to 
form one or more different compounds. In aqueous solutions these compounds 
are called 'hydrates'; for non-aqueous systems the term 'solvate' is sometimes 
used. Two types of compound can be considered: one can coexist in stable 
equilibrium with a liquid of the same composition, and the other cannot behave 
in this manner. In the former case the compound is said to have a congruent 
melting point; in the latter, to have an incongruent melting point. 

Figure 4.5 illustrates the phase reactions in the manganese nitrate-water 
system. Curve AB is the freezing point curve. The solubility curve BCDEFG 
for Mn(N03) 2 in water is not continuous owing to the formation of several 
different hydrates. The area above curve ABCDEFG represents homogeneous 
liquid solutions. Mixtures of the hexahydrate and solution exist in areas BCH 
and ICD. The tetrahydrate is the stable phase in region DEJ and the dihydrate 
in EKF. The rectangular areas under FH, IJ and KL represent completely 

40 - 

20 - 



2 ° 


£ -20 

40 - 




Solution QJ 





A / 


Ice + solution \ / ~2 




O ! 

CM 1 

X | 
CM ' 
+ 1 



<«■ 1 


Ice + Mn (N0 3 ) 2 -6H 2 

1,1 1 

1 1 

0.2 04 0.6 0.8 

Mass fraction Mn(N0 3 ) 2 


Figure 4.5. Phase diagram for the system Mn(N0 3 ) 2 -H 2 

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Phase equilibria 143 

solidified systems (ice and hexahydrate, hexa- and tetrahydrates, and tetra- 
and dihydrates, respectively). Point B is a eutectic or cryohydric point with the 
co-ordinates — 36 °C and 0.405 mass fraction of Mn(N03) 2 . 

Point C in Figure 4.5 indicates the melting point (25.8 °C) and composition 
(0.624 mass fraction) of the hexahydrate. Thus when a solution of this com- 
position is cooled to 25.8 °C it solidifies to the hexahydrate, i.e. no change in 
composition occurs. Point C, therefore, is a congruent point. Similarly, point E 
is the congruent point for the tetrahydrate (melting point 37.1 °C, composition 
0.713). Points D and Fare the other two eutectic points of the system. Point G is 
the transition point at which the dihydrates decomposes into the monohydrate 
and water, i.e. it is the incongruent melting point of the dihydrate. The vertical 
broken line at 0.834 mass fraction represents the composition of the dihydrate. 

The behaviour of manganese nitrate solutions on cooling can be traced in the 
same manner as that described above for simple eutectic systems. The solution 
concentrations and the proportions of solid and solution can similarly be 
deduced graphically. The process of isothermal evaporation in congruent melt- 
ing systems presents an interesting phenomenon. For example, the mixture 
represented by point X in Figure 4.5 represents a slurry of ice and solution; 
but when sufficient water is removed to bring the system composition into the 
region to the right of curve AB, it becomes a homogeneous liquid solution. 
When more water is removed, so that region BCG is entered, the system 
partially solidifies again, depositing crystals of the hexahydrate. On further 
evaporation, once the composition exceeds 62.4 per cent of Mn(N03) 2 , e.g. at 
point Y, the system solidifies completely to a mixture of the hexa- and tetra- 
hydrates. The reverse order of behaviour occurs on isothermal hydration. 

The formation of eutectics and solvates with congruent points is observed in 
many organic, aqueous inorganic and metallic systems. The case illustrated 
above is a rather simple example. Some systems form a large number of solvates 
and their phase diagrams can become rather complex. Ferric chloride, for 
example, forms four hydrates, and the FeC^-H^O phase diagram exhibits 
five-cryohydric points and four congruent points. 

A solvate that is unstable in the presence of a liquid of the same composition 
is said to have an incongruent melting point. Such a solvate melts to form a 
solution and another compound, which may or may not be a solvate. For 
instance, the hydrate Na 2 S04 • IOH2O melts at 32.4 °C to give a saturated 
solution of sodium sulphate containing a suspension of the anhydrous salt; 
hence, this temperature is the incongruent melting point of the decahydrate. 
The terms 'meritectic point' and 'transition point' are also used instead of the 
expression 'incongruent melting point'. 

Figure 4.6 illustrates the behaviour of the system sodium chloride-water. The 
various areas are marked on the diagram. AB is the freezing point curve and 
BC is the solubility curve for the dihydrate. Point B (—21 °C) is a eutectic or 
cryohydric point at which a solid mixture of ice and NaCl • 2H 2 of fixed 
composition (0.29 mass fraction of NaCl) is deposited. At point C (0.15 °C) the 
dihydrate decomposes into the anhydrous salt and water; this is, therefore, the 
incongruent melting point, or transition point, of NaCl • 2H 2 0. The vertical 
line commencing at 0.619 mass fraction of NaCl represents the composition of 

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°\ /-- 

Solution + 

A c 







Solution \ / NaCL-2H,0 
+ \ 5 / 
ice \J 



NaCl-2H 2 +ice 

1 1 1 1 ! ! 


01 0.2 03 0.4 0.5 06 0.7 

Mqss fraction NaCL 
Figure 4.6. Phase diagram for the system NaCl-H20 

the dihydrate. If this system had a congruent melting point, which it does not 
have, this line would meet the peak of the extension of curve BC (e.g. see Figure 

Many aqueous and organic systems exhibit eutectic and incongruent points. 
Several cases are known of an inverted solubility effect after the transition point 
(see Figure 3.1b); the systems Na2SC>4-H20 and Na2CC>3-H20 are particularly 
well-known examples of this behaviour. 

Salt hydrates for energy storage 

There has been a growing interest in recent years in the use of salt hydrates as 
heat storage materials, e.g. solar heat for space-heating purposes or in small 
heat parks for personal uses. The hydrates are melted in the energy absorbing 
stage and they subsequently release heat at the phase transition temperature 
when they recrystallize. Ideally the hydrates should have a congruent melting 
point so that the phase transition crystal ^ melt ^ crystal can be repeated 
indefinitely. In practice, however, many otherwise acceptable hydrates exhibit 
slightly incongruent behaviour and have to be used in admixture with other 

Examples of hydrates that have been considered for domestic application 
include CaCl 2 • 6H 2 0, Na 2 S0 4 • 10H 2 O, Na 2 S 2 3 • 5H 2 0, Na 2 HP0 4 • 12H 2 
and CH 3 COONa • 3H 2 (Kimura, 1980; Gronvold and Meisingset, 1982; Feil- 
chenfeld and Sarig, 1985; Kimura and Kai, 1985; Tamme, 1987). 

4.3.3 Solid solutions 

Many binary systems when submitted to a cooling operation do not at any 
stage deposit one of the components in the pure state: both components are 
deposited simultaneously. The deposited solid phase is, in fact, a solid solution. 

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Phase equilibria 




2 ioo 

| 90 





Solid soln 
+ liquid 


yr\ ^ 

i i i i 




I i ; 



-2 90 

f 80 

"0 2 0.4 0.6 0.8 1.0 

Mass fraction ^-naphthol 



Solid soln 
+ liquid. 


i ; i i 


2 0.4 6 0.8 1.0 

Mass fraction /?-naphthylamine 

Figure 4.7. Solid solutions: (a) continuous series (naphthalene-f3-naphthol; (b) minimum 
melting point (naphthalene-j3-naphthylamine) 

Only two phases can exist in such a system: a homogeneous liquid solution and 
a solid solution. Therefore, from the reduced phase rule, F' = 1, so an invariant 
system cannot result. One of three possible types of equilibrium diagram can be 
exhibited by systems of this kind. In the first type, illustrated in Figure 4.7a, all 
mixtures of the two components have freezing or melting points intermediate 
between the melting points of the pure components. In the second type shown 
in Figure 4.7b, a minimum is produced in the freezing and melting point curves. 
In the third, rare, type of diagram, a maximum is exhibited in the curves. 

Figure 4.7a shows the temperature-concentration phase diagram for the 
system naphthalene-/3-naphthol, which forms a continuous series of solid 
solutions. The melting points of pure naphthalene and /3-naphthol are 80 and 
120 °C, respectively. The upper curve is the liquidus or freezing point curve, the 
lower the solidus or melting point curve. Any system represented by a point 
above the liquidus is completely molten, and any point below the solidus 
represents a completely solidified mass. A point within the area enclosed by 
the liquidus and solidus curves indicates an equilibrium mixture of liquid and 
solid solution. Point X, for instance, denotes a liquid of composition L in 
equilibrium with a solid solution of composition S, and point Y a liquid L' in 
equilibrium with a solid 5". 

The phase reactions occurring on the cooling of a given mixture can be traced 
as follows. If a homogeneous liquid represented by point A (60 per cent 
/3-naphthol) is cooled slowly, it starts to crystallize when point L (105 °C) is 
reached. The composition of the first crystals is given by point S (82 per cent /3- 
naphthol). As the temperature is lowered further, more crystals are deposited 
but their composition changes successively along curve SS', and the liquid 
composition changes along curve LL'. When the temperature is reduced to 
94 °C (points L' and S'), the system solidifies completely. The over-all composi- 
tion of the solid system at some temperature represented by, say, point A' is the 
same as that of the original homogeneous melt, assuming that no crystals have 

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146 Crystallization 

been removed during the cooling process, but the system is no longer homo- 
geneous because of the successive depositions of crystals of varying composi- 
tion. The changes occurring when a solid mixture A' is heated can be traced in 
a manner similar to the cooling operation. 

Figure 4.7b shows the relatively uncommon, but not rare, type of binary 
system in which a common minimum temperature is reached by both the upper 
liquidus and lower solidus curves. These two curves approach and touch at 
point M. The example shown in Figure 4.7b is the system naphthalene-/?- 
naphthylamine. Freezing and melting points of mixtures of this system do not 
necessarily lie between the melting points of the pure components. Three sharp 
melting points are observed: 80 °C (pure naphthalene), 1 10 °C (pure /3-naphthy- 
lamine) and 72.5 °C (mixture M, 0.3 mass fraction /3-naphfhylamine). Although 
the solid solution deposited at point M has a definite composition, it is not 
a chemical compound. The components of such a minimum melting point 
mixture are rarely, if ever, present in stoichiometric proportions. Point M, 
therefore, is not a eutectic point: the liquidus curve is completely continuous; 
it only approaches and touches the solidus at M. The phase reactions occurring 
when mixtures of this system are cooled can be traced in the same manner 
as that described for the continuous series solid solutions. 

4.4 Enthalpy-composition diagrams 

The heat effects accompanying a crystallization operation may be determined 
by making heat balances over the system, although many calculations may be 
necessary, involving knowledge of specific heat capacities, heats of crystalliza- 
tion, heats of dilution, heats of vaporization, and so on. Much of the calcula- 
tion burden can be eased, however, by the use of a graphical technique in which 
enthalpy data, solubilities and phase equilibria are represented on an enthalpy- 
composition {H—x) diagram, sometimes known as a Merkel chart. 

The use of the H—x diagram for the analysis of chemical engineering unit 
operations such as distillation, evaporation and refrigeration processes, is 
now quite common, and the procedures are well described in textbooks, e.g. 
Coulson and Richardson (1991), McCabe, Smith and Harriott (1985). These 
charts are less frequently applied to crystallization processes, however, because 
not many H—x diagrams are available. 

Among the few enthalpy-composition charts for solid-liquid systems pub- 
lished in the open literature (all for aqueous solutions) are: 

ammonium nitrate (Othmer and Frohlich, 1960) 
calcium chloride (Hougen, Watson and Ragatz, 1943) 
calcium nitrate (Scholle and Brunclikova, 1968) 
magnesium sulphate (McCabe, 1935) 

sodium tetraborate (borax) (Scholle and Szmigielska, 1965) 
sodium carbonate (Tyner, 1955) 
sodium hydroxide (McCabe, 1935) 
sodium sulphate (Foust et al., 1960) 

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Phase equilibria 




Xa Kc Xg 

Concentration, x 
Figure 4.8. An adiabatic mixing process represented on an H-x diagram 

boric acid (Scholle and Szmigielska, 1965) 

hydrazine (Tyner, 1955) 

urea (Banerjee and Doraiswamy, 1960) 

The construction of an H-x diagram is laborious (McCabe, 1935) and would 
normally be undertaken only if many calculations were to be performed, e.g. on 
a system of commercial importance. Nevertheless, once an H-x chart is avail- 
able its use is simple, and a great deal of information can be obtained rapidly. 
If the concentration x of one component of a binary mixture is expressed as 
a mass fraction, the enthalpy is expressed as a number of heat units per unit 
mass of mixture, e.g. Btulb -1 or Jkg~'. Molar units are less frequently used 
in crystallizer design practice. 

The basic rule governing the use of an H—x chart is that an adiabatic mixing, 
or separation, process is represented by a straight line. In Figure 4.8 points A 
and B represent the concentrations and enthalpies xa, Ha and x B , H B of two 
mixtures of the same system. If A is mixed adiabatically with B, the enthalpy 
and concentration of the resulting mixture is given by point C on the straight 
line AB. The exact location of point C, which depends on the masses wia and m B 
of the two initial mixtures, can be determined by the mixture rule or lever-arm 

m A (x c - x A ) = m B (x B - x c ) 




m B x B - 
m A 

■m A x A 
-m B 


Similarly, if mixture A were to be removed adiabatically from mixture C, the 
enthalpy and composition of residue B can be located on the straight line 
through points A and C by means of the equation 

x B 

m c x c - m A x A 
m c -m A 


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H-x charts in SI units for aqueous solutions of sodium carbonate and 
sodium sulphate (both recalculated from original data) are given in Figures 
4.9 and 4.10, respectively, and a chart for magnesium sulphate, retained in its 
original Imperial units, is given in Figure 4.11. 

In Figure 4.11, for example, the isotherms in the region above curve pabcdq 
represent enthalpies and concentrations of unsaturated aqueous solutions of 
MgS04, and the very slight curvature of these isotherms indicates that the heat 
of dilution of MgS04 solutions is very small. Point p (zero enthalpy) represents 
pure water at 32 °F, point n the enthalpy of pure ice at the same temperature. 
The portion of the diagram below curve pabcdq, which represents liquid-solid 
systems, can be divided into five polythermal regions: 





Na,C0 3 iOH 2 



No 2 C0 3 7H 2 Na 2 C0 3 H 2 

1.2 0.4 0.6 0.8 

Concentration -weight fraction of Na,C0 3 

Figure 4.9. Enthalpy-concentration diagram for the system Na2C03-H20 

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Phase equilibria 




















Na 2 S0 4 
NajS0 4 IOH.,0 


0.1 0.2 0.3 0.4 0.5 

Concentration-weight fraction of Na 2 S0 4 

Figure 4.10. Enthalpy-concentration diagram for the system Na2S04-H20 

pae solutions of MgS04 in equilibrium with pure ice 

abfg equilibrium mixtures of MgS04 • I2H2O and saturated solution 

bcih equilibrium mixtures of MgSC>4 • 7H2O and saturated solution 

cdlj equilibrium mixtures of MgS04 • 6H2O and saturated solution 

dqrk equilibrium mixtures of MgSC>4 • H2O and saturated solution 

In between these five regions lie four isothermal triangular areas, which 
represent the following conditions: 

aef{25°¥) mixtures of ice, cryohydrate a and MgSC>4 • I2H2O 

bfh (37.5 °F) mixtures of solid MgS0 4 • 12H 2 and MgS0 4 • 7H 2 in a 

21 per cent MgSC^ solution 
cji (118.8 °F) mixtures of solid MgS0 4 • 7H 2 and MgS0 4 • 6H 2 in a 

33 per cent MgSCU solution 
dkl (154.4 °F) mixtures of solid MgS0 4 • 6H 2 and MgS0 4 • H 2 in a 

37 per cent MgSCU solution 

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O.IO 0.20 0.30 40 0.50 

Concentration, weight fraction MgS0 4 

Figure 4.11. Enthalpy-concentration diagram for the 
McCabe, 1963, by courtesy of McGraw-Hill) 

system MgS04-H20. (From 

The short vertical lines fg and ih represent the compositions of solid 
MgS0 4 • 12H 2 (0.359 mass fraction MgS0 4 ) and MgS0 4 • 7H 2 (0.49 mass 
fraction). The following example demonstrates the use of Figure 4.11. 


Calculate (a) the quantity of heat to be removed and (b) the theoretical crystal 
yield when 50001b of a 30 per cent solution of MgS04 by mass at 110°F is 
cooled to 70 °F. Evaporation and radiation losses may be neglected. 

Figure 4.12 indicates the relevant section 
Figure 4.11. 

not to scale - of the H—x diagram in 

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Phase equilibria 


4. , 





1 I 

1 1 



1 1 
1 1 

x 5 1 x A- t x 9 



0.26 0.30 


Mass fraction of Mg S0 4 

Figure 4.12. Graphical solution of Example (enlarged portion of Figure 4.11 - not to scale) 

(a) Initial solution, A 
Cooled system, B 
Enthalpy change 
Heat to be removed 

x A = 0.30, H A 

x B = 0.30, H B 


44 x 5000 

-220 000 Btu 

(b) The cooled system B, located in the region bcih in Figure 4.11, com- 
prises MgS04 • 7H2O crystals in equilibrium with solution S on curve be. The 
actual proportions of solid and solution can be calculated by the mixture rule. 

Solution composition x$ = 0.26 

Crystalline phase composition xc = 0.49 

4.5 Phase change detection 

4.5.1 Thermal analysis 

A phase reaction is always accompanied by an enthalpy change (section 2.12), 
and this heat effect can readily be observed if a cooling curve is plotted for the 
system. In many cases a very simple apparatus can be used. A large glass test- 
tube, fitted with a stirrer and a thermometer graduated in increments of 0.1 °C 
and held in a temperature-controlled environment, will often suffice. The 
temperature of the system is recorded at regular intervals of say 1 min. 

A smooth cooling curve is followed until a phase reaction takes place, when 
the accompanying heat effect causes an arrest or change in slope. Figure 4.13a 
shows a typical example for a pure substance. AB is the cooling curve for the 
homogeneous liquid phase. At point B the substance starts to freeze and the 
system remains at constant temperature, the freezing point, until solidification 
is complete at point C. The solid then cools at a rate indicated by curve CD. It is 
possible, of course, for the liquid phase to cool below the freezing point, and 

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Figure 4.13. Some typical cooling curves 

some systems may withstand appreciable degrees of supercooling. The dotted 
curve in Figure 4.13a denotes the sort of path followed if supercooling occurs. 
Seeding of the system will minimize these effects. 

Figure 4.13b shows the type of cooling curve obtained for a binary system in 
which eutectic or compound formation occurs. The temperature of the homo- 
geneous liquid phase falls steadily along curve EF until, at point F, deposition 
of the solid phase commences. The rate of cooling changes along curve FG as 
more solid is deposited. The composition of the remaining solution changes 
until the composition of the eutectic is reached, then crystallization or freezing 
continues at constant temperature (line GH), i.e. the eutectic behaves as a single 
pure substance. The completely solidified system cools along curve HL. Super- 
cooling, denoted by the dotted lines, may be encountered at both arrest points 
if the system is not seeded. 

Figure 4.13c shows a typical cooling curve for a binary mixture that forms 
a series of solid solutions. The first arrest, K, in the curve corresponds to the 
onset of freezing, and this represents a point on the liquidus. The second arrest, 
L, occurs on the completion of freezing and represents a point on the solidus. 
It will be noted that no constant-temperature freezing point occurs in such 
a system. 

The discontinuities may not always be clearly defined on a cooling curve 
(temperature 6 versus time t plot). In such cases, the arrest points can often be 
greatly exaggerated by plotting an inverse rate curve (8 versus dt/d6, i.e. the 
inverse of the cooling rate). A typical plot is shown in Figure 4.14. 

Equilibria in solid solutions are better studied by a heating than by a cooling 
process. This is the basis of the thaw-melt method. An intimate mixture of 
known composition of the two pure components is prepared by melting, 
solidifying and then crushing to a fine powder. A small sample of the powder 
is placed in a melting-point tube, attached close to the bulb of a thermometer 
graduated in increments of 0. 1 °C, and immersed in a stirred bath. The tem- 
perature is raised slowly and regularly at a rate of about 1 °C in 5min. The 
'thaw point' is the temperature at which liquid first appears in the tube; this is 
a point on the solidus. The 'melt point' is the temperature at which the last solid 
particle melts; this is a point on the liquidus. Only pure substances and eutectic 
mixtures have sharp melting points. The thaw-melt method is particularly 

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Phase equilibria 


Inverse cooling rote, d//d9 

Figure 4.14. Detection of the arrest point for a single substance: (a) on a temperature-time 
curve, (b) on an inverse rate curve 

A Composition — 

A Composition -— 

Figure 4.15. Construction of equilibrium diagrams from 'thaw-melf data: (a) eutectic 
system; (b) solid solution 

useful if the system is prone to supercooling, and it has the added advantage of 
requiring only small quantities of test material. 

The construction of equilibrium diagrams from cooling or thaw-melt data is 
indicated in Figure 4.15. In practice, however, a large number of different 
mixtures of the two components A and B, covering the complete range from 
pure A to pure B, would be tested. The liquidus curves are drawn through the 
first-arrest points, the solidus curves through the second-arrest points. Only 
at 100 per cent A, 100 per cent B and the eutectic point do the liquidus and 
solidus meet. 

Differential thermal analysis (DTA) 

Differential thermal analysis is a method used for observing phase changes and 
measuring the associated changes in enthalpy. A small test sample, often only 
a few milligrams, is heated in close proximity to a sample of reference material 
in an identical container. The reference material, chosen for its similarity to the 
test sample, must not exhibit any phase change over the temperature range 
under consideration. 

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Reference jf 
somple v // 

Melting / / 
point / /\ 






Reference temperature 


Figure 4.16. Differential thermal analysis: (a) comparative heating curves, (b) differential 
temperature curve for a single substance 

When the test sample undergoes a phase change, there will be heat release or 
absorption. For example, if it melts it will absorb heat and its temperature will 
lag behind that of the reference material {Figure 4.16a). The difference in 
temperature between the two samples is detected by a pair of matched thermo- 
couples and recorded as a function of time. The area between the differential 
curve (Figure 4.16b) and the base line is a function of the enthalpy associated 
with the phase change. 

Differential scanning calorimetry (DSC) 

Differential scanning calorimetry is another calorimetric technique for observ- 
ing solid-liquid phase changes. Two independently controlled heaters allow 
the sample and reference pans to be heated at a fixed rate. The instrument 
detects the temperature difference AT between the sample and reference, dur- 
ing heating or cooling, and records the amount of heat added to or removed 
from the sample at the sample temperature to compensate for the temperature 
difference. The melting point and enthalpy of fusion of the sample material can 
thus be determined simultaneously from the DSC curve. An exothermic reac- 
tion in the sample results in a positive peak in the DSC curve. An endothermic 
reaction gives a negative peak. 

Some typical DSC curves are shown in Figure 4.17. The negative peaks 
indicate endothermic melting. The height of the peak quantifies the enthalpy 
of fusion. A pure sample gives a sharp peak (Figure 4.17d) while an impure 
sample would show a broader peak, an indefinite start and a blunt maximum. 
Different types of DSC curve will be obtained for different types of phase 
equilibria. For example, Figure 4.17b indicates, for a binary system, the beha- 
viour for a simple eutectic and Figure 4.17c shows the behaviour for the 
formation of a series of solid solutions. 

A good introductory account of the basic principles and practical require- 
ments of a range of modern techniques of thermal analysis is given by Brown 
(1988). The development of a differential scanning calorimeter, coupled with 
a personal computer, for the measurement of solid-liquid equilibria, has been 
described by Matsuoka and Ozawa (1989). 

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Phase equilibria 155 


Figure 4.17. DSC curves: (a) pure component, (b) eutectic mixture, (c) solid solution 
(T m melting point, T e eutectic point, Tq onset temperature, T p peak temperature) 

4.5.2 Dilatometry 

The dilatometric methods for detecting phase changes utilize volume changes in 
the same way as the calorimetric methods utilize thermal effects. Dilatometry is 
widely used in the analysis of melts and particularly of fats and waxes (Bailey, 
1950; Swern, 1979). The techniques and equipment are usually quite simple. 

Solids absorb heat on melting and, with the notable exception of ice, expand. 
They evolve heat when they undergo polymorphic transformation to a more 
stable polymorphic and contract. Consequently, dilatometric (specific volume- 
temperature) curves bear a close resemblance to calorimetric (enthalpy- 
temperature) curves. The melting dilation corresponds to the heat of fusion, 
and the coefficient of cubical expansion, a, corresponds to the specific heat 
capacity, c. The ratio c/a is virtually a constant independent of temperature. 

A dilatometer used for fats and waxes is shown in Figure 4.18. Mercury, or 
some other suitable liquid, is used as the confining fluid and the liquid thread in 

Filling device 


5 cm 


Figure 4.18. Gravimetric dilatometer and filling device. (After Bailey, 1950) 

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156 Crystallization 

the capillary, C, communicates with the reservoir, R. Volume changes in the 
sample, S, are measured by weighing the liquid in the reservoir before and after. 
The small expansion bulb, B, is warmed to expel any air that enters the end of 
the capillary when the flask is detached. Owing to the high density of mercury 
and the accuracy with which weighings can be made, volume changes as small 
as 10~ 5 cm 3 g _1 have been detected. 

Melting points can be determined with great precision by dilatometry. A plot 
of dilation versus temperature usually gives two straight lines - one for the solid 
dilation, which generally has a steep slope, and one for the liquid, with a low 
slope. The point of intersection of these two lines give the melting point, which 
may often be estimated to ±0.01 °C. 

4.6 Three-component systems 

4.6.1 Construction of ternary diagrams 

The phase equilibria in ternary systems can be affected by four variables, viz. 
temperature, pressure and the concentration of any two of the three compon- 
ents. This fact can be deduced from the phase rule: 

F+F = 3+2 

which indicates that a one-phase ternary system will have four degrees of 
freedom. It is impossible to represent the effects of the four possible variables 
in a ternary system on a two-dimensional graph. For solid-liquid systems, 
however, the pressure variable may be neglected, and the effect of temperature 
will be considered later. 

The composition of a ternary system can be represented graphically on a 
triangular diagram. Two methods are in common use. The first utilizes the 
equilateral triangle, and the method of construction is shown in Figure 4.19a. 
The apexes of the triangle represent the pure components A, B and C. A point 
on a side of the triangle stands for a binary system, AB, BC or AC; a point 
within the triangle represents a ternary system ABC. The scales may be con- 
structed in any convenient units, e.g. weight or mole percent, weight or mole 
fraction, etc., and any point on the diagram must satisfy the equation 
^4 + 5+C=lor 100. The quantities of the components A, B and Cin a given 
mixture M are represented by the perpendicular distance from the sides of the 

Special triangular graph paper is required if the equilateral diagram is to be 
used, and for this reason many workers prefer to employ the right-angled 
triangular diagram which can be drawn on ordinary squared graph paper. The 
construction of the right-angled isosceles triangle is shown in Figure 4.19b. 
Again, as in the case of the equilateral triangle, each apex represents a pure 
component A, B or C, a point on a side a binary system, and a point within the 
triangle a ternary system; in all cases A + B + C = 1 or 100. The quantities 
of A, B and C in a given mixture M are represented by the perpendicular 

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Phase equilibria 




o A, 0.8 5, 0.2 C 

d0.2 A, 0.6 B, 0.2 C 

7 0.2 A, 0.2 5, 0.6 C 

A 0.8 A, 0.2 3, 0C 













0.2 04 6 08 10 

A scale -«- 


0.2 0.4 0.6 0.8 1.0 
A scale — - 

Figure 4.19. Construction of equilateral triangular diagrams 

distances to the sides of the triangle. If two compositions, A and B, B and C, or 
A and C are known, the composition of the third component is fixed on both 
triangular diagrams. 

Two actual plots are shown in Figure 4.19c to illustrate the interpretation of 
these diagrams. For clarity the C scale has been omitted from the right-angled 
diagram; the C values can be obtained from the expression C = I — (A + B). 
The 'mixture rule' is also illustrated in Figure 4.19c. When any two mixtures X 
and Y are mixed together, the composition of the final mixture Z is represented 
by a point on the diagram located on a straight line drawn between the points 
representing the initial mixtures. The position of Z is located by the expression 

mass of mixture X distance YZ 

mass of mixture Y distance XZ 

For example, if one part of a mixture X (0.1v4, 0.5S, 0.4C) is mixed with one 
part of a mixture Y(0.5A, 0.35, 0.2C), the composition of the final mixture Z 
(0.3 A, 0.4B, 0.3C) is found on the line XY where XZ = YZ. Again, if 3 parts of 
Fare mixed with 1 part of X, the mixture composition Z' (0.4^4, 0.355, 0.25C) 
is found on the line XY where XZ' = 3(FZ'). The mixture rule also applies to 
the removal of one or more constituents from a system. Thus, one part of 
a mixture X removed from 2 parts of a mixture Z would yield one part of a 
mixture Y given by: 

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158 Crystallization 

mass of original Z 
mass of X removed 



Similarly, one part of X removed from 4 parts of Z' would yield 3 parts of 
a mixture Y given by 

mass of original Z' YX 4 
mass of X removed YZ' 1 

The principle of the mixture rule is the same as that employed in the opera- 
tion of lever-arm problems, i.e. m\l\ = mih, where m is a mass and / is the 
distance between the line of action of the mass and the fulcrum. For this reason, 
the mixture rule is often referred to as the lever-arm or centre of gravity 

Although ternary equilibrium data are most frequently plotted on equilateral 
diagrams, the use of the right-angled diagram has several advantages. Apart 
from the fact that special graph paper is not required, it is claimed that 
information may be plotted more rapidly on it, and some people find it easier 
to read. In this section the conventional equilateral diagram will mostly be 
employed, but one or two illustrations of the use of the right-angled diagram 
will be given. 

4.6.2 Eutectic formation 

Equilibrium relationships in three-component systems can be represented on 
a temperature-concentration space model as shown in Figure 4.20. The ternary 
system ortho-, meta- and /?ara-nitrophenol, in which no compound formation 
occurs, is chosen for illustration purposes. The three components will be 
referred to as O, M and P, respectively. Points O' , M' and P on the vertical 





MO A~ 

(45°C) (3I.5 Q C) 



Figure 4.20. Eutectic formation in the three-component system o-, m- and p-nitrophenol: 
(a) temperature-concentration space model; (h) projection on a triangular diagram 

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Phase equilibria 159 

edges of the model represent the melting points of the pure components ortho- 
(45 °C), meta- (97 °C) and para- (114°C). The vertical faces of the prism 
represent the temperature-concentration diagrams for the three binary systems 
O—M, OP and M-P. These diagrams are each similar to that shown in Figure 
4.4 described in the section on binary eutectic systems. In this case, however, 
the solidus lines have been omitted for clarity. 

The binary eutectics are represented by points A (31.5 °C; 72.5 per cent O, 
27.5 per cent M), B (33.5 °C; 75.5 per cent O, 24.5 per cent M) and C (61.5 °C; 
54.8 per cent M, 45.2 per cent P). Curve AD within the prism represents the 
effect of the addition of the component P to the O—M binary eutectic A. 
Similarly, curves BD and CD denote the lowering of the freezing points of the 
binary eutectics B and C, respectively, on the addition of the third component. 
Point D, which indicates the lowest temperature at which solid and liquid 
phases can coexist in equilibrium in this system, is a ternary eutectic point 
(21.5 °C; 57.7 per cent O, 23.2 per cent M, 19.1 per cent P). At this temperature 
and concentration the liquid freezes invariantly to form a solid mixture of the 
three components. The section of the space model above the freezing point 
surfaces formed by the liquidus curves represents the homogeneous liquid 
phase. The section below these surfaces down to a temperature represented 
by point D denotes solid and liquid phases in equilibium. Below this temper- 
ature the section of the model represents a completely solidified system. 

Figure 4.20b is the projection of the curves AD, BD and CD in Figure 4.20a 
on to the triangular base. The apexes of the triangle represent the pure compon- 
ents O, M and P and their melting points. Points A, B and C on the sides of 
the triangle indicate the three binary eutectic points, point D the ternary 
eutectic point. The projection diagram is divided by curves AD, BD and CD 
into three regions which denote the three liquidus surfaces in the space model. 
The temperature falls from the apexes and sides of the triangle towards the 
eutectic point D, and several isotherms showing points on the liquidus surfaces 
are drawn on the diagrams. The phase reactions occurring when a given ternary 
mixture is cooled can now be traced. 

A molten mixture with a composition as in point X starts to solidify when 
the temperature is reduced to 80 °C. Point X lies in the region ADCM, so 
pure meta- is deposited on decreasing temperature. The composition of the 
remaining melt changes along line MXX' in the direction away from point M 
representing the deposited solid phase (the mixture rule). At X', where line 
MXX' meets curve CD, the temperature is about 50 °C, and at this point 
a second component (para-) also starts to crystallize out. On further cooling, 
meta- and para- are deposited and the liquid phase composition changes in 
the direction X'D. When melt composition and temperature reach point D, 
the third component (ortho-) crystallizes out, and the system solidifies with- 
out any further change in composition. A similar reasoning may be applied 
to the cooling, or melting, of systems represented by points in the other 
regions of the diagrams. 

An example of the use of a ternary eutectic diagram for the assessment 
of a melt recrystallization process (for nitrotoluene isomers) is given in 
section 8.2.1. 

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160 Crystallization 

4.6.3 Two salts and water 

There are many different types of phase behaviour encountered in ternary 
systems consisting of water and two solid solutes. Only a few of the simpler 
cases will be considered here; attention will be devoted to a brief survey of 
systems in which there is (a) no chemical reaction, (b) formation of a solvate, 
e.g. a hydrate, (c) formation of a double salt, and (d) formation of a hydrated 
double salt. 

At one given temperature the composition of, and phase equilibria in, a 
ternary aqueous solution can be represented on an isothermal triangular dia- 
gram. The construction of these diagrams has already been described. Poly- 
thermal diagrams can also be constructed, but in the case of complex systems 
the charts tend to become congested and rather difficult to interpret. 

No compound formed 

This simplest case is illustrated in Figure 4.21 for the system KN03-NaN03- 
H2O at 50 °C. Neither salt forms a hydrate, nor do they combine chemically. 
Point A represents the solubility of KNO3 in water at the stated temperature 
(46.2g/100g of solution) and point C the solubility of NaN0 3 (53.2g/100g). 
Curve AB indicates the composition of saturated ternary solutions that are 
in equilibrium with solid KNO3, curve BC those in equilibrium with solid 
NaNC>3. The upper area enclosed by ABC represents the region of unsaturated 
homogeneous solutions. The three 'triangular' areas are constructed by draw- 
ing straight lines from point B to the two apexes of the triangle; the composi- 
tions of the phases within these regions are marked on the diagram. At point B 
the solution is saturated with respect to both KNO3 and NaN03, and from the 
reduced phase rule F' = 1. This means that point B, generally referred to as 
a eutonic point, is univariant, i.e. invariant when the temperature is fixed. 

The effect of isothermal evaporation on such a system can be shown as 
follows. If water is evaporated from an unsaturated solution represented by 


NaN0 3 

*t X s 

Figure 4.21. Phase diagram for the system KN0 3 -NaN0 3 -H 2 at 50 °C 

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Phase equilibria 


point X\ in the diagram, the solution concentration will increase, following line 
X\X2- Pure KNO3 will be deposited when the concentration reaches point Xi. If 
more water is evaporated to give a system of composition X3, the composition 
of the solution will be represented by point X' 3 on the saturation curve AB: and 
when composition X\ is reached, by point B: any further removal of water will 
cause the deposition of NaNC^ as well as KNO3. All solutions in contact with 
solid will thereafter have a constant composition B. For this reason the eutonic 
point B is sometimes referred to as the drying-up point of the system. After the 
complete evaporation of water the composition of the solid residue is indicated 
by point X5 on the base line. 

Similarly, if an unsaturated solution, represented by a point located to the 
right of B in the diagram, were evaporated isothermally, only NaNCh would be 
deposited until the solution composition reached the drying-up point B, when 
KNO3 would also be deposited. The solution composition would thereafter 
remain constant until evaporation was completed. If water is removed isotherm- 
ally from a solution of composition B, the composition of the deposited solid 
is given by point X(, on the base line, and it remains unchanged throughout the 
remainder of the evaporation process. 

The effect of the addition of one of the salts to the system KN03-NaN03- 
H2O at 50 °C is shown in Figure 4.22a. This time the equilibria are plotted on 
a right-angled triangular diagram simply to demonstrate the use of this type 
of chart. Points A and C, as in Figure 4.21, refer to the solubilities at 50 °C of 
KNO3 and NaNC>3, respectively. Curves AB and BC indicate the saturated 
ternary solutions in equilibrium with solid KNO3 or NaNC>3, and show, for 
instance, that the solubility of KNO3 in water is depressed when NaN03 is 
present in the system, and vice versa. 

Take, for example, a binary system NaNC^-H^O represented by point Y\ 
(0.7 mass fraction of NaNC>3 and 0.3 H2O). As this point lies in the 'triangular' 


0.2 4 0.6 0.8 

Mass fraction NaN03 

0.2 0.4 6 0.8 

Mass fraction NaN03 

Figure 4.22. Phase diagrams for the system KN0 3 -NaN0 3 -H 2 0: (a) at 50 °C; (h) at 24 
and 100 °C 

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162 Crystallization 

region to the right of curve BC, the system consists of a saturated solution of 
NaNC>3, with a composition given by point C, and excess solid NaNC>3. If 
a quantity of KNO3 is added to this binary system, the temperature being 
kept constant at 50 °C so that the new composition is represented by point Y 2 
(0.64 NaN0 3 , 0.1 KNO3, 0.26 H 2 0), the composition of the ternary saturated 
solution in contact with the excess solid NaN0 3 present is given by Y' 2 (0.46 
NaN0 3 , 0.15 KNO3, 0.39 H 2 0) on the line drawn from the apex N through Y 2 
to meet curve BC. As more KNO3 is added, the solution concentration alters, 
following curve CB. At point B the solution becomes saturated with respect to 
both NaN0 3 and KN0 3 ; its concentration is 0.4 NaN0 3 , 0.29 KNO3, 0.31 
H2O. If after this point further quantities of KNO3 are added to bring the 
system concentration up to some point Y3, no more KNO3 dissolves, the 
solution composition remains at point B. 

The interpretation of these phase diagrams is aided by remembering the rule 
of mixtures - i.e. on the removal or addition of any component from or to a 
system, the composition of the system changes along a straight line drawn from 
the original composition point to the apex representing the pure given com- 
ponent. In Figure 4.22a the right-angled apex represents pure water, the top 
apex K pure KNO3 and the other acute apex N pure NaNC>3. 

The effect of temperature on the system KN03-NaN03-H20 is shown in 
Figure 4.22b. Two isotherms, A'B'C and A"B"C", for 25 and 100 °C, 
respectively, are drawn on this diagram. The lower left-hand area enclosed by 
A'B'C represents homogeneous unsaturated solutions at 25 °C, the larger area 
enclosed by A" B"C" unsaturated solutions at 100 °C. The line B'B" shows the 
locus of the drying-up points between 25 and 100 °C. To illustrate the effect of 
temperature changes in the system, let point Z\ refer to the composition (0.5 
NaN03, 0.1 KNO3, 0.4 H 2 0) of a certain quantity of the ternary mixture. 
From the position of Z\ in the diagram it can be seen that at 100 °C the system 
would be a homogeneous unsaturated solution, but at 25 °C it would consist 
of pure undissolved NaNC>3 in a saturated aqueous solution of NaNC>3 and 
KNO3. Thus pure NaNC>3 would crystallize out of the solution Z\ on cooling 
from, say, 100 to 25 °C, in fact at about 50 °C. Despite the phase changes, of 
course, the overall system composition remains at Z\ until one or more com- 
ponents are removed. At 25 °C the composition of the solution in contact with 
the crystals of NaN0 3 is given by the intersection of the line from N through Z\ 
with curve B'C, i.e. at point Z\ (0.43 NaN0 3 , 0.11 KNO3, 0.46 H 2 0). 
The quantity of NaNC>3 which would crystallize out at 25 °C is given by the 
mixture rule 

mass of crystals deposited length Z\Z\ 
mass of saturated solution length Z\N 

where N represents the NaNC>3 apex of the triangle. 

When a 'pure' solute is to be crystallized from a ternary two-solute system by 
cooling, there is usually a temperature limit below which the desired solute 
becomes 'contaminated' with the other solute. This can be demonstrated by con- 
sidering a system represented by point Z 2 in Figure 4.22b. The composition at 
Z 2 is 0.3 NaN0 3 , 0.45 KN0 3 , 0.25 H 2 0; at 100 °C the system is a homogeneous 

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Phase equilibria 


unsaturated solution. At 25 °C, however, this point lies in the region where both 
solid NaNCh and KNO3 are in equilibrium with a saturated solution of both 
salts, its composition being given by point B'. If it is desired to cool solution Z2 
in order to yield only KNO3 crystals, then the temperature limitation is found 
by drawing a straight line from the KNO3 apex K through point Z2 and 
producing it to meet the drying-up line B'B" at Z' 2 . Point Z' 2 occupies the 
position of an invariant point on an isotherm; by referring to Figure 4.22a it can 
be seen that it corresponds approximately to point B on the 50 °C isotherm. 
Thus solution Z2 must not be cooled below 50 °C if only KNO3 crystals are to 
be deposited. 

Solvate formation 

When one of the solutes in a ternary system is capable of forming a compound, 
with the solvent, the phase diagram will contain more regions to consider than 
in the simple case described above. A common example of solvate formation is 
the production of a hydrated salt in a ternary aqueous system. Figure 4.23 
shows the isothermal diagrams for the system NaCl-Na2S04-H20 at two 
temperatures, 17.5 and 25 °C, at which different phase equilibria are exhibited. 
Sodium sulphate combines with water, under certain conditions, to form 
Na2SC>4 • IOH2O. Sodium chloride, however, does not form a hydrate at the 
temperature being considered. Figure 4.23a shows the case where the decahy- 
drate is stable in the presence of NaCl, and Figure 4.23b that of the decahydrate 
being dehydrated by the NaCl under certain conditions. 

Points A and C in Figure 4.23a represent the solubilities of NaCl (26.5 mass 
per cent) and Na2S04 (13.8 per cent) in water at 17.5 °C, curves AB and BC the 
ternary solutions in equilibrium with solid NaCl and Na2S04 • IOH2O, respect- 
ively. Point D shows the composition of the hydrate Na2S04 • IOH2O. For 
convenience, the following symbols are used on the diagram to mark the phase 


Na 2 S0 4 NaCl 

5+/y+so 4 


No 2 S0 4 



Figure 4.23. Phase diagrams for the system NaCl-Na 2 S0 4 -H20: (a) at 17.5 °C; (b) at 

25 °C 

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regions: S = solution; H = hydrate Na 2 S0 4 • 10H 2 O; S0 4 = Na 2 S0 4 and 
CI = NaCl. The solution above curve ABC is unsaturated. The lowest triangu- 
lar region represents a solid mixture of Na 2 S0 4 , Na 2 SC>4 • 10H 2 O and NaCl. 
Point B is the eutonic or drying-up point of the system. 

In Figure 4.23b, points A and D denote the solubilities of NaCl (26.6 mass per 
cent) and Na 2 S0 4 (21.6 per cent) in water at 25 °C, point E the composition of 
Na 2 S0 4 • 10H 2 O. In this diagram there are three curves, AB, BC and CD, 
which give the composition of the ternary solutions in equilibrium with NaCl, 
Na 2 S0 4 and Na 2 S0 4 • 10H 2 O. The various phase regions are indicated on the 
diagram. If NaCl is added to a system in the region CDE, i.e. to an equilibrium 
mixture of solid Na 2 S0 4 • 10H 2 O in a solution of NaCl and Na 2 S0 4 , the 
solution concentration will change along curve DC. When point C is reached, 
the NaCl can only dissolve by dehydrating the Na 2 S0 4 • 10H 2 O, and anhy- 
drous Na 2 S0 4 is deposited. Further addition of NaCl will result in the complete 
removal of the decahydrate from the system, the solution concentration follow- 
ing curve CB; under these conditions the excess solid phase consists of anhy- 
drous Na 2 S0 4 . At the eutonic point B the solution is saturated with respect to 
both NaCl and Na 2 S0 4 . 

The effects of isothermal evaporation, salt additions and cooling can be 
traced from Figure 4.23 in a manner similar to that outlined for Figures 4.21 
and 4.22. 

Double salt formation 

Cases are encountered in ternary systems where the two dissolved solutes 
combine in fixed proportions to form a definite double compound. Figure 
4.24 shows two possible cases for a hypothetical aqueous solution of two salts 
A and B. Point C on the AB side of each triangle represents the composition of 
the double salt; points L and O show the solubilities of salts A and B in water at 
the given temperature. Curves LM and NO denote ternary solutions saturated 

(a) (b) 

Figure 4.24. Formation of a double salt: (a) stable in water; (b) decomposed by water 

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Phase equilibria 


with salts A and B, respectively, curve MN ternary solutions in equilibrium with 
the double salt C. The significance of the various areas is marked on the 

The isothermal dehydration of solutions in Figure 4.24a can be traced in the 
manner described for Figures 4.21 and 4.22. Point M is the eutonic or drying-up 
point for solutions located to the left of broken line WR, point TV that for 
solutions to the right of this line. A solution on line WM behaves as a solution 
of a single salt in water; when its composition reaches point M, a mixture of 
salt A and double salt C crystallizes out in the fixed ratio of the lengths PCjAP. 
Similarly, a solution on line WN yields a mixture of B and C, in the ratio 
CQ/QB, when its composition reaches point N. A solution represented by a point 
on line WR also behaves as a solution of a single salt; when its composition 
reaches point R, the double compound C crystallizes out and neither of salts A 
and B is deposited at any stage. Point R, therefore, is the third drying-up point 
of the system. An example of this type of system is ammonium and silver 
nitrates in water, giving the double salt NH4NO3 • AgNOj. 

The phase diagram in Figure 4.24b shows a different case. There are only two 
drying-up points, M and N, in this system, the first for solutions located to the 
left, the second for solutions to the right of line WN. Each solution on lines 
WM and WN behaves as a solution of single salt in water. The line WC does 
not cross the saturation curve MN of the double salt but cuts the saturation 
curve for salt B, indicating that the double salt is not stable in water; it is 
decomposed and salt B is deposited. An example of this type of system is 
glaserite, a non-stoichiometric double salt of potassium and sodium sulphates 
with the formula K^Na^Oz^. 

Hydmted double salt 

Figure 4.25a shows the phase diagram for the case of a hydrated double salt 
that is stable in water. The best-known examples of this type of system are the 

Figure 4.25. Formation of a hydrated double salt: (a) stable in water, (b) decomposed by 

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166 Crystallization 

alums (M\S0 4 • M 2 n (S0 4 ) 3 • 24H 2 0, where M l and M m represent mono- and 
tervalent cations, e.g. M l = Na, K, NH4, Cs, Rb, Tl or hydroxylamine; 
M HI = Al, Ti, V, Cr, Mn, Fe, Co or Ga; the sulphate radical may be replaced 
by selenate) and the Tutton salts (M I 2 M u (S0 4 ) 2 ■ 6H 2 0, where M l and M u 
represent mono- and bivalent ions, respectively, e.g. M 1 = NH4, K, Rb, Cs or 
Te; M m = Ni, Mn, Mg, Fe, Co, Zn or Cu). 

In the case depicted salt A forms a hydrate of composition H. Its saturation 
curve is LM . Salt B is anhydrous and its saturation curve is ON. Point W 
represents water. Salts A and B combine together to form a hydrated double 
salt of composition denoted by point C within the triangular diagram. MN is 
the saturation curve for the hydrated double salt. The compositions of the 
phases in the eight separate regions are indicated in the diagram. The only 
region in which the pure hydrated double salt will crystallize out of solution, at 
the temperature for which the particular phase diagram is drawn, is the area 
bounded by MNC. 

In Figure 4.25a line WC cuts the saturation curve MN of the hydrated 
double salt, which indicates that the salt is stable in the presence of water. In 
Figure 4.25b line WC does not cross curve MN, which indicates that the 
hydrated double salt decomposes in the presence of water. This is a comparat- 
ively rare behaviour, but an example is the case of MgS04 • Na 2 SC>4 • 4H 2 
(astrakanite) at 25 °C. 

4.6.4 Solid solutions 

Ternary systems comprising water and two electrolytes containing a common 
ion often yield solid solutions. Such a system can be represented in the manner 
indicated in Figure 4.26: an isothermal diagram for salts A and B and solvent 
water W. Points a and b represent the solubilities of salts A and B at the given 

Figure 4.26. Solid solution formation in a ternary system, e.g. two salts in water 

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Phase equilibria 


temperature. The curve ab represents the equilibrium solubilities of mixtures 
of salts A and B. The sector ab W represents unsaturated solutions, and sector 
AabB represents mixture of AB solid solutions (crystals) in equilibrium with 
aqueous solutions saturated with salts A and B. The broken tie lines connect 
equilibrium mixtures of liquid solutions (curve ab) and solid solutions (base 
line AB). 

For example, a solution S, isothermally evaporated to a condition repres- 
ented by point M would yield a mixture of crystals (solid solution) of overall 
composition C at one end of the tie line suspended in a solution of composition 
represented by point L at the other end. If evaporation were to be continued to 
dryness, the overall composition of the solid solution would be represented by 
point F on line AB. The deposited crystals would not be homogeneous, how- 
ever, since they would have successively grown from a whole range of solution 
compositions and would tend to reflect these conditions by their outer layers 
being of slightly different composition from their insides. 

An alternative method of representing ternary solid solution systems graph- 
ically is to plot the concentration of one component in the solid phase against its 
composition in the liquid phase. On this basis, Roozeboom in 1891 showed that 
five different types of system were possible. Only two of these will be mentioned 
here, however, but a good account of all five types of behaviour is given by 
Blasdale (1927). 

Type I behaviour is characterized by complete miscibility, with the concen- 
tration of one of the salts in the liquid phase exceeding that in the solid phase 
for all concentrations (Figure 4.27a). Examples of type I systems include 
K 2 S04-(NH4) 2 S04-H 2 0, KH 2 As0 4 -H 2 and K-alum-NH 4 -alum-H 2 0. 

Type II behaviour is also characterized by complete miscibility, but while the 
concentration of one of the salts in the liquid phase exceeds that in the solid 
phase for a certain rate of concentrations, it is less for the remaining concen- 
trations. In other words, at one particular concentration the A : B salt ratios in 
the solid and liquid phases are identical (Figure 4.27b). An example of this less- 
common type II behaviour is the system KCl-KBr-H 2 0. 

Mass fraction of A in crystals 
(a) (b) 

Figure 4.27. Solid solution formation in a ternary system: (a) Type 1; (b) Type 2 

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168 Crystallization 

Solid solutions with a eutectic 

Not all solid solution systems form continuous series; some exhibit partial 
miscibility, i.e. one solid solution being partially miscible with another, and, 
as in the case of partially miscible liquids, the phase region between the two 
homogeneous phases is referred to as the miscibility gap. Partially miscible 
solid solution systems can exhibit a number of different types of behaviour, but 
only one simple case will be described here for illustration purposes. 

Figure 4.28 shows an example of two solid solutions that form a eutectic, 
a fairly common occurrence in organic melts. Curve AE indicates how increas- 
ing amounts of component B lower the freezing point of AB liquid mixtures. 
Curve BE shows the effect of component A on B. All systems above curve AEB 
(the liquidus) are homogeneous liquid and all systems below curve ACEDB (the 
solidus) are solid. In the sectors to the left of A CF and to the right of BDG, the 
solid phases are homogeneous solid solutions a and f3, respectively. The sector 
FCEDG, the miscibility gap, encloses heterogeneous mixtures of the two solid 
solutions a and (3. The sectors ACE and BDE contain mixtures of a + liquid 
and /3 + liquid respectively. Point E represents the temperature and composi- 
tion of the eutectic, a conglomerate of solid solutions a and (3. The cooling of 
a liquid mixture X to some temperature Y may be traced as follows. Point Y, 
which lies in sector BDE, represents a suspension of solid solution j3, of 
composition S, in equilibrium with a liquid of composition L. The proportion 
of solid to liquid is represented by the distance ratio LY: SY (the mixture rule). 

When the miscibility gap extends close to the pure component compositions, 
it can be difficult to distinguish between this type of system and that of the 
simple eutectic described in section 4.3.1. The problem of terminal solid 
solutions is discussed in section 7.2. 







' 1 


L / 


S / 


a J 


\ P 

1 F 

6 \ 

(Mass fraction of component B) 

Figure 4.28. Two solid solutions that form a eutectic 

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Phase equilibria 169 

4.6.5 Equilibrium determinations 

For multicomponent systems the composition of the equilibrium solid phase 
can be determined indirectly by the so-called wet-residues method, first pro- 
posed by Schreinemakers (1893), in which the need for solid-liquid separation 
before analysis is avoided. In practice, the equilibrium system is allowed to 
settle and then most of the saturated supernatant solution is decanted off the 
sedimented solids. A sample of the wet solids is then scooped out and quickly 
weighed in a closed weighing bottle, to avoid solvent loss, and subsequently 
analysed by the most convenient analytical technique. 

The method of wet residues is based on the application of the straight-line 
mixture rule on a phase diagram. For a ternary system, the solid-liquid phase 
equilibria can be represented on a triangular diagram, with the equilibrium 
solution composition being represented by a point on the solubility curve and 
the wet-residue composition by a point within the triangle. By virtue of the 
properties of a triangular phase diagram, the three points representing the 
compositions of equilibrium solubility, wet-residue and the equilibrium solid 
phase, must lie on a straight line (section 4.6. 1). The point at which a line drawn 
through the solubility and wet-residue points and extended to meet the side of 
the triangle therefore gives the composition of the equilibrium solid phase. 

Although extrapolations are commonly made graphically on phase dia- 
grams, algebraic extrapolation is less subjective, more accurate, and lends itself 
to the application of statistical methods which minimize errors. Mathematical 
extrapolation procedures for the method of wet residues have been described by 
Ricci (1966) and Schott (1961). 

The synthetic complex method of solid-liquid equilibrium determination in 
multicomponent systems offers an alternative procedure to that of the wet- 
residue method, and is capable of yielding more rapid results. The procedure is 
as follows. Several mixtures of the solutes are prepared, covering a range of 
compositions, and known amounts of solvent are added to each sample. Thus a 
number of 'synthetic complexes' of known composition are obtained and their 
composition points can be plotted within a phase diagram. The samples are 
then shaken or agitated to equilibrate at constant temperature, using any 
convenient method, after which the clear supernatant saturated solution is 
analysed. Again, as in the wet-residues method, a line is drawn through the 
'solution' point, its corresponding 'complex' point, and then extended to one 
side of the phase diagram (triangular for a ternary system) to give the composi- 
tion of the solid phase. 

Purdon and Slater (1946) give good accounts of the practical difficulties that 
may be encountered in applying both the wet-residues and synthetic complex 
methods of solid phase analysis. 

4.7 Four-component systems 

A one-phase, four-component or quaternary system has five degrees of free- 
dom. Therefore the phase equilibria in these systems may be affected by the five 

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variables: pressure, temperature and the concentrations of any three of the four 
components. To represent quaternary systems graphically, one or more of the 
above variables must be excluded. The effect of pressure on solid-liquid sys- 
tems may be ignored, and if only one temperature is considered an isothermal 
space model can be constructed. If the concentration of one of the components 
is excluded, usually the liquid solvent, a two-dimensional graph can be drawn, 
but this simplification will be described later. 

4.7.1 Three salts and water 

The first, simple, type of quaternary system to be considered here consists of 
three solid solutes, A, B and C, and a liquid solvent, S. No chemical reaction 
takes place between any of the components, e.g. water and three salts with 
a common ion. The isothermal space model for this type of system can be 
constructed in the form of a tetrahedron {Figure 4.29a) with the solvent at the 
top apex and the three solid solutes on the base triangle. The four triangular 
faces of the tetrahedron represent the four ternary systems A-B-C, A—B—S, 
A-C-S and B—C—S. The three faces, excluding the base, have the appearance 
of the 'two salts and water' diagram shown in Figure 4.21. 

A point on an edge of the tetrahedron represents a binary system, a point 
within it a quaternary. On the faces ABS, BCS and ACS the solubility curves 
meet at points L, M and N, respectively, which represent the solvent saturated 
with two solutes. They are the starting points for the three curves LO, MO and 
NO, which denote solutions of three solutes in the solvent; point O represents 
the solution which, at the given temperature, is saturated with respect to all 
three solutes. All these curves form three curved surfaces within the space 
model. The section between these surfaces and the apex of the tetrahedron 
indicates unsaturated solution, that between the surfaces and the triangular 
base complex mixtures of liquid and solid. 

Figure 14.29b shows another way in which systems of this type can be 
represented as a space model. Here it takes the form of a triangular prism 
where the apexes of the triangular base represent the three solid components 

Figure 4.29. Isothermal representation of a quaternary system of the 'three salts with a 
common ion in water' type: (a) tetrahedral space model; (h) triangular prism space model; 
(c) Jdnecke projection 

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Phase equilibria 111 

and the vertical scale the liquid solvent. The interpretation of this model is 
similar to that just described for the tetrahedron; the same symbols have been 

For a complete picture of the phase behaviour of quaternary systems a space 
model is essential; yet, because of its time-consuming construction, a two- 
dimensional 'projection' is frequently employed. Such a projection, named after 
E. Janecke, is shown in Figure 4.29c. In this type of isothermal diagram the 
solvent is excluded. The curved surfaces A'LON, B'MOL and C'NOM in 
Figures 4.29a and 4.29b, which represent solutions in equilibrium with solutes 
A, B and C, respectively, are projected on to the triangular base and become 
areas ALON, BMOL and CNOM in Figure 4.29c. Curves LO, MO and NO 
denote solutions in equilibrium with two solutes, viz. A and B, B and C, A and 
C, respectively, while point O represents a solution in equilibrium with the three 
solutes. For this type of system the projection diagram can be plotted in terms 
of mass or mole fractions or percentages. 

4.7.2 Reciprocal salt pairs 

The second, and more important, type of quaternary system that will be 
considered is one consisting of two solutes and a liquid solvent where the two 
solutes inter-react and undergo double decomposition (metathesis). This beha- 
viour is frequently encountered in aqueous solutions of two salts that do not 
have a common ion. Typical examples of double decomposition reactions of 
commercial importance are 

KC1 + NaN0 3 ^ NaCl + KNO3 

NaN0 3 + i(NH 4 ) 2 S0 4 ^ NH 4 N0 3 + ±Na 2 S0 4 

KC1 + ±Na 2 S0 4 ^ NaCl + ±K 2 S0 4 

NaCl + i(NH 4 ) 2 S0 4 ^ NH 4 C1 + iNa 2 S0 4 
NaN0 3 + iK 2 S0 4 ^ KN0 3 + iNa 2 S0 4 

The four salts in each of the above systems form what is known as a 
'reciprocal salt pair'. Although all four may be present in aqueous solution, 
the composition of any mixture can be expressed in terms of three salts and 
water. Thus, from the phase rule point of view, an aqueous reciprocal salt pair 
system is considered to be a four-component system. 

Reciprocal salt pair solutions may be represented on an isothermal space 
model, in the form of either a square-based pyramid or a square prism. Figure 
4.30a indicates the pyramidal model: the four equilateral triangular faces 
stand for the four ternary systems AX-AY- W, AY-BY-W, BY-BX-W and 
4X-BX-W {W = water) for the salt pair represented by the equation 

AX + BY ^ AY + BX 

The apex of the pyramid denotes pure water, its base the anhydrous quaternary 
system AX- A Y-BX-BY. Points L, M, N and O on the four triangular faces of 

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Figure 4.30. Isothermal representation of a quaternary system of the 'reciprocal salt paif 
type: (a) square-based pyramid; (h) square prism 

the pyramid indicate the equilibria between two salts and water. Point P, which 
represents a solution of three salts AX, BX and BY in water saturated with all 
three salts, is a quaternary invariant point. So is Q, which shows the equilib- 
rium between salts AX, AY and BY and water. Curves OP, NP and LQ, MQ, 
which join these quaternary invariant points P and Q to the corresponding 
ternary invariant points on the triangular faces of the pyramid, represent 
solutions of three salts in water saturated with two salts, and so does curve 
PQ, joining the two quaternary invariant points. 

The square-prism space model (Figure 4.30b) illustrates another way in which 
a quaternary system of the reciprocal salt pair type may be represented. The 
vertical axis stands for the water content, and the points on the diagram are the 
same as those marked on Figure 4.30a. In both diagrams all surfaces formed 
between the internal curves represent solutions of three salts in water saturated 
with one salt, all internal curves solutions of three salts in water saturated with 
two salts, and the two points P and Q solutions of three salts in water saturated 
with the three salts. The section above the internal curved surfaces denotes 
unsaturated solutions, the section below them mixtures of liquid and solid. 

4.7.3 Janecke diagrams 

In order to simplify the interpretation of the phase equilibria in reciprocal salt 
pair systems, the water content may be excluded. The curves of the space model 
can then be projected on to the square base to give a two-dimensional graph, 
called a Janecke diagram as described in section 4.7.1. A typical projection is 
shown in Figure 4.31a; the lettering is that used in Figure 4.30. The enclosed 
areas, which represent saturation surfaces, indicate solutions in equilibrium 
with one salt, the curves solutions in equilibrium with two salts, points P and 
Q solutions in equilibrium with three salts. 

Molar or ionic bases must be used in this type of diagram for reciprocal salt 
pairs. The four corners of the square represent 100 mol of the pure salts AX, 
BX, BY and AY. Any point inside the square denotes 100 mol of a mixture of 
these salts; its composition can always be expressed in terms of three salts. The 

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Phase equilibria 





Soln 1 

+ / 








o\ — 


/ Soln 

/ + 

/ AY 





— Jf scale 

BX 80 60 40 20 

Figure 4.31. Interpretation of the Jdnecke diagram for reciprocal salt pairs: (a) projection 
of the surfaces of saturation on to the base; (b) method of plotting 

scales in Figure 4.31b are marked in ionic percentages of A, B, X and Y. Take, 
for example, lOOmol of a mixture expressed as 


moles of 


of ions 























The totals of the A + B ions (e.g. the basic radicals) and the X + Y ions (e.g. the 
acidic radicals) must always equal 100. Thus point a, indicating this mixture 
can be plotted: the square is divided by the two diagonals into four right-angled 
triangles, and point a lies in triangles AX .AY . BX and AX .AY .BY. There- 
fore the composition of the above mixture could also have been expressed in 
terms of salts ^Z (40 mol), ^4 7(40mol)and BY (20mol). In a similar manner, it 
can be shown that point (3 which lies within the two triangles AX . BX . BY and 
BX .BY . AY represents 100 mol of a mixture with a composition expressed 
either by 50 BY, 30 AX and 20 BX, or by 50 BX, 30 AY and 20 BY. 

Although it is usually more convenient to plot ionic percentages on the 
square, it is quite in order to plot mole percentages of the salts direct. The 
numerical scales marked on Figure 4.31b must now be ignored. If point a is 
considered to lie in triangle AX .AY . BX, representing a mixture 20 AX, 60 A Y 
and 20 BX, the compositions of the two salts at opposite ends of the diagonal 
A Y and BX are used for plotting purposes. Thus point a is located by 60 along 
the horizontal A Y scale and 20 up the vertical BX scale. If a is taken to lie in 
triangle AX .AY . BY, the composition is represented by 40 AX, 20 BY, 40 A Y, 

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174 Crystallization 





/ 1 

A /, 


Pi ' / 



/ / J 





\/ / \ 


__ z 


I'/ \ 



\ / / ' 


\ ' 











Figure 4.32. Jdnecke projections for aqueous solutions of a reciprocal salt pair, showing 
(a) two congruent points, (b) congruent and incongruent points 

and the A Y and BX compositions are used for plotting. A similar reasoning may 
be applied to the plotting of point (3 in triangles AX.BX.BY and AY. BY .BX. 

Figure 4.32 shows Janecke diagrams for solutions of a given reciprocal salt 
pair at different temperatures. These two simple cases will be used to demon- 
strate some of the phase reactions that can be encountered in such systems. 
Both diagrams are divided by the saturation curves into four areas which are 
actually the projections of the surfaces of saturation (e.g., see Figure 4.32b). 
Salts AX and BY can coexist in solution in stable equilibrium: the solutions are 
given by points along curve FQ. Salts BX and AY, however, cannot coexist in 
solution because their saturation surfaces are separated from each other by 
curve FQ. Thus AX and BY are called the stable salt pair, or the compatible 
salts, BX and A Y the unstable salt pair, or the incompatible salts. In Figure 4.32a 
the AX-BY diagonal cuts curve FQ which joins the two quarternary invariant 
points, while in Figure 4.32b curve P'Q' is not cut by either diagonal. These are 
two different cases to consider. 

Point P represents a solution saturated with salts AX, BX and BY, Q one 
saturated with salts AX, BY and AY. In Figure 4.32a both P and Q lie in their 
'correct' triangles, i.e. AX .BX .BY and AX .BY .AY, respectively, and solu- 
tions represented by P and Q are said to be congruently saturated. In Figure 
4.32b point Q' lies in its 'correct' triangle, AX . BY .AY, but P' lies in the 
'wrong' triangle, the same as Q'. Point Q', therefore, is congruent and point 
P' is incongruent. 

Isothermal evaporation 

The phase reactions occurring on the removal of water from a reciprocal salt 
pair system will first be described with reference to Figure 4.32a. Point a which 
lies on the BY saturation surface represents a solution saturated with salt BY. 
When water is removed isothermally from this solution, the pure salt BY is 

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Phase equilibria 175 

deposited and the solution composition (i.e. the composition of the salts in 
solution, the water content being ignored) moves from a towards d along the 
straight line drawn from BY through a to meet curve QM. When a sufficient 
quantity of water has been removed, the solution composition reaches point a' 
and here the solution is saturated with two salts, BY and AY, 

Further evaporation results in the deposition of AY as well as BY; the 
composition of the solid phase being deposited is given by point M. The overall 
composition of deposited solid therefore moves from BY towards a" on the line 
BY .AY. The solution composition, being depleted in solid M, moves away 
from point M towards Q. On reaching point Q, three salts AX, A Y and BY are 
deposited. The composition of the solid phase deposited is also given by point 
Q; the overall composition of the solid phase, assuming that none has been 
removed from the system, by point a". The solution composition, the water 
content being ignored, and the composition of the deposited solid phase remain 
constant at point Q for the rest of the evaporation process, and the overall 
solids content changes along line a" a, composition a representing the com- 
pletely dry complex. Point Q is a quaternary drying-up point for all solutions 
represented by points within triangle AX .AY .BY. 

The isothermal evaporation of solution b on the diagonal can be traced as 
follows. If point b lies on the saturation surface, it represents a solution 
saturated with salt BY. While salt BY is being deposited, the solution composi- 
tion changes along the diagonal from b towards b' . At b' the solution becomes 
saturated with salts AX and BY. This ternary system (AX-BY-H2O) thereafter 
dries up, without change in composition, at point b' . Point b', therefore, is a 
ternary drying-up point. 

If point c lies on the saturation surface, it represents a solution saturated with 
salt AX. When this solution is evaporated isothermally, AX is deposited and the 
solution composition changes along line cc'. At d salt BY also crystallizes out 
and the composition of the solid phase deposited is given by b', the point at 
which the diagonal crosses line PQ. The solution composition, therefore, 
changes along line c'P, and at P the three salts AX, BY and BX are co- 
deposited: point P is the quaternary drying-up point for all solutions repres- 
ented by points within triangle AX . BX . BY. 

The isothermal evaporation of a solution denoted by point w in Figure 4.32b 
can be traced in the same manner as that described for point a in Figure 4.32a. 
Q' is the drying-up point. The evaporation of solution x can be traced as 
follows. At x the solution is saturated with salt BY, and this salt is deposited 
until the solution composition reaches x', where the solution is saturated with 
the two salts AX and BY. The composition of the solid phase being deposited at 
this stage is given by point R on the diagonal. As evaporation proceeds, the 
solution composition changes from point x' along line x'Q', i.e. in a direction 
away from point R, and at Q' the solution is saturated with the three salts AX, 
AY and BY. Both solution and deposited solids thereafter have a constant 
composition until evaporation is complete: Q' is the quaternary drying-up 

Point Q' is also the drying-up point for a solution represented by point y. The 
solution composition changes along line yy' while salt AX crystallizes out, and 

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176 Crystallization 

then from y' towards P' while the two salts AX and BY of composition 0' are 
deposited. At P' the solution is saturated with the three salts AX, BX and BY, 
the composition of the solid phase deposited at this point being given by R. On 
further evaporation, the solution composition remains constant at P' while salts 
AX and BY are deposited and salt BX is dissolved. When all BX has dissolved, 
the solution composition changes from P' towards Q', and the solution finally 
dries up at Q'. 

Point P 1 , therefore, is incongruent. It is not a true drying-up point except for 
the case where the original complex lies within the triangle representing the 
three salts of which it is the saturation point, i.e. AX, BY and BX. Point z may 
be taken as an example of this case. On evaporation, the solution composition 
changes from z to z' while salt AX is deposited, from z' towards P' while salts 
AX and BX are deposited. The composition of the solid phase at this latter 
stage is given by point O'. At P 1 this solution is saturated with salts AX, BX and 
BY. Further evaporation results in the deposition of AX and BY and the 
dissolution of BX. The solution dries up at point P' . 

Representation of water content 

So far in the discussion of Janecke projections for reciprocal salt pair systems 
the water content has been ignored. This is not too serious, because much 
information can be obtained from the projection before consideration of the 
quantity of water present. One way in which the water content can be repres- 
ented is shown in Figure 4.33a; the plan shows the projection of the saturation 
surfaces, the elevation indicates the water contents. To avoid unnecessary 
complication, the elevation only shows the horizontal view of the particular 
saturation curve concerned in the problem. 

The isothermal evaporation of water from a complex a was considered in 
Figure 4.32a, where point a, representing the composition of the given complex, 
was taken to lie on the saturation surface. In Figure 4.33a the isothermal 
dehydration of an unsaturated solution S is considered, the dissolved salt 
having the same composition a as that in Figure 4.32a. Point S, therefore, is 
located on the elevation vertically above point a in the plan. The exact position 
of S is determined by the water content of the given solution, i.e. distance Saj, 
on the water scale denotes the moles of water per lOOmol of salt content. Line 
5«3, called the water line, represents the course of the isothermal dehydration. 
Points Q and M are similarly located on the elevation, according to their 
corresponding water contents, vertically above points Q and M on the plan. 
Point a' lies on curve QM vertically above a' in the plan. Point T on the 
elevation represents the water content of a saturated solution of pure salt BY, 
the salt to be deposited. 

Three construction lines can now be drawn on the elevation. Line Td cuts the 
water line at point a. The BY saturation surface is assumed for simplicity to be 
plane, so Tad is a line on this surface. The Y corner of the elevation represents 
pure salts A Y and BY and all their mixtures. The line drawn from d to Y (BY 
on plan) cuts the water line at a\, that from Q to Y (a" on plan) at a-i. 

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Phase equilibria 1 77 


Cold c b 







/ / 

/ // 



/ y 

__ 1 _/' 

v // 



\ / / ._ 



0/ 5' 





Figure 4.33. Representation of water content: (a) isothermal evaporation; (b) crystal- 
lization by cooling 

When water is removed isothermally from the unsaturated solution S, 
the water content falls along the water line Saj. When point a is reached, the 
solution is saturated with salt BY, and pure BY starts to crystallize out. The 
quantity of water to be removed to achieve this condition is determined from 
the water scale readings on the elevation diagram, i.e. Sa mol of water has to be 
removed from a system containing 100 mol of salts dissolved in Sa^, mol of 
water. Salt BY is deposited while the water content falls from a to fli, and at 
point a\ the solution (of composition a') becomes saturated with salts BY and 
A Y. Both salts are deposited while the water content falls from a\ to ai, and the 
overall deposited solids content changes along line BY '/a" on the plan. At point 
«2 the solution (composition Q) is saturated with respect to the three salts AX, 
AY and BY, and further evaporation from a^ to a^ proceeds at constant 
solution composition Q. The solids composition changes along line a"a on 
the plan. 

Crystallization by cooling 

The graphical procedure described above, viz. the drawing of a plan and 
elevation, provides a simple pictorial representation of the phase reactions 

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178 Crystallization 

occurring in a given system at two different temperatures. Figure 4.33b shows 
two isotherms labelled 'hot' and 'cold', respectively; they are in fact the curves 
from Figure 4.32, plotted on one diagram, and the same lettering is used. By 
way of example, two different cooling operations will be considered. 

Point a on curve Q'M' represents a hot solution saturated with the two salts 
AY and BY. When it is cooled to the lower temperature point a lies in the BY 
field of the projection. Line BY/ a is drawn on the plan to meet curve QM at b, 
but point b represents the solution composition only if point a lies on the BY 
saturation surface in the 'cold' projection, i.e. if pure BY was crystallizing out. 
To find the actual solution composition and the composition of the deposited 
solid phase, point b is projected from the plan onto curve QM in the elevation. 

Point Y on the elevation diagram represents salts A Y or BY or any mixture of 
them. Line Ya is drawn on the elevation and then produced to meet curve QM 
at c. It can be seen that in this case points b and c do not coincide. This means 
that the deposited solid phase is not pure salt BY but some mixture of BY and 
A Y. Point c is projected from the elevation onto the plan, and line cadi?, drawn. 
Thus the final solution composition is given by point c, and the overall solid 
phase composition by point d. 

If pure salt BY was required to be produced during the cooling operation, the 
water content of the system would have to be adjusted accordingly. Solution 
point c has to move to become coincident with point d, and solid point d has to 
move to BY on the plan. In this case, therefore, water has to be added to the 
system, e.g. to the hot solution before cooling. The quantity of water required 
per 100 mol of salts is given by the vertical distance ae on the elevation. 

A different sequence of operations is shown in another section of Figure 
4.33b. Point w on curve F'Q' represents a solution saturated with salts AX and 
BY at the higher temperature. At the lower temperature, however, point w lies 
in the BY field of the diagram. If the correct amount of water is present in the 
system, pure BY crystallizes out on cooling, and the solution composition is 
given by point x located on line BY/w produced to meet curve FQ. A cyclic 
process can now be planned. 

The pure salt BY is filtered off and a quantity of solid mixture, e.g. of 
composition z, is added to solution x. The quantity of solid z to be added, 
calculated by the mixture rule, must be the amount necessary to give complex y, 
the composition of which is chosen so that, on being heated to the higher 
temperature, it lies in the AX field, yields the original solution w and deposits 
the pure salt AX. Thus the sequence of operations is 

1. Cool solution w to the lower temperature 

2. Filter off solid BY 

3. Add solid mixture z to the mother liquor x to give complex y 

4. Heat the complex to the higher temperature 

5. Filter off solid AX 

6. Cool mother liquor w to the lower temperature, and so on 

Of course, the water contents at each stage in the cycle must be adjusted so that 
the solutions deposit only one pure salt at a time. The quantities of water to be 

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Phase equilibria 179 

added or removed can be estimated graphically on the elevation diagram in the 
manner described above for solution a. 

Only the simplest type of reciprocal salt pair diagram has been considered 
here. Many systems form hydrates or double salts: in others the stable salt pair 
at one temperature may become the unstable pair at another. For information 
on these more complicated systems reference should be made to specialized 
works on the phase rule. The monographs by Blasdale (1927) and Purdon and 
Slater (1946) are particularly noteworthy in this respect; graphical solutions of 
problems of commercial importance are given, and five-component aqueous 
systems are analysed. Teeple (1929) and Fitch (1970) also give accounts of the 
use of multicomponent phase diagrams for the design of industrial fractional 
crystallization processes. 

The phase diagrams described in this section are by no means limited to 'salts 
in water' systems, as a comparison between Figures 4.20 and 4.29 will clearly 
show. A worked example of the use of a diagram similar to both Figures 4.20b 
and 4.29c is given in section 8.2.1 to demonstrate the recovery of one pure 
component from a ternary organic eutectic system by cooling melt crystal- 
lization. The use of multicomponent phase diagrams for selecting appropriate 
crystallization methods for a wide range of separation procedures including 
cooling, evaporating, salting-out, adduct formation, etc. with both organic and 
inorganic systems has been extensively demonstrated by Chang and Ng (1998), 
Cesar and Ng (1999) and Wibowo and Ng (2000). 

4.8 'Dynamic' phase diagrams 

One of the problems of trying to establish reliable phase equilibria in multi- 
component solid-liquid systems is that very long periods of contact between 
crystals and solution are often necessary before the equilibrium state is 
approached. In fact, some systems can appear to be unable to achieve a stable 
equilibrium, in which case a meaningful phase diagram cannot be constructed. 

Not only are reliable multicomponent phase equilibria difficult to measure in 
the laboratory, the measured data may be found to be inapplicable to certain 
industrial procedures where, for example, contact times between solid and 
liquid phases can be quite short and true equilibrium state conditions are not 

It has long been appreciated that phase equilibria of complex salt systems 
measured under laboratory conditions may have limited industrial use. It was 
first noted by van't Hoff (1903), when crystallizing salts from seawater, that 
certain thermodynamically expected stable salts never crystallized. Even exceed- 
ingly slow crystallization together with deliberate seeding by the salts them- 
selves did not help. Yet the salts in question all occupied clearly defined zones 
on the appropriate stable phase diagrams. 

Studies on similar systems were made in the USSR by Kurnakov in the 1920s 
and Valyashka in the 1940s (see Hadzeriga, 1967), in Germany by Autenrieth 
(1953) and in the USA by Hadzeriga (1967). Attempts to reproduce in the 
laboratory conditions of natural saline lake evaporation appropriate to the 

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180 Crystallization 

MgCI 2 
MgCI 2 .6H 2 

MgS0 4 .H 2 
MgS0 4 .6H 2 

MgS0 4 .7H 2 

Na 2 S0 4 



MgCI 2 .KCI.6H 2 

MgS0 4 .KCI.3H 2 


KoCI 5 




Figure 4.34. A 'dynamic' phase diagram superimposed on a conventional equilibrium 
diagram for the quinary system Na, K, Mg, CI, SO4 in water at 25 °C 

solar evaporation of natural brines have led to proposals for the use of 
'dynamic' as opposed to conventional 'equilibrium' phase diagrams. Figure 
4.34 is such a diagram for the quinary aqueous system Na + , K + , Mg 2+ , Cl~ 
and SC>4~ saturated with NaCl, using a Janecke projection. The particular area 
of interest represents only the upper third of the full phase diagram. 

Although there is a large central field of kainite (MgS04 • KC1 • 3H2O) on the 
'equilibrium' diagram (zones bounded by bold lines in Figure 4.34a) kainite 
does not crystallize out when brines in this region are evaporated in solar 
ponds. In fact, under these operating conditions, all the phase boundaries are 
changed; the sylvite (KC1) field, for example, is slightly enlarged; carnallite 
(KC1 • MgS0 4 • 6H 2 0) and hexahydrate (MgS0 4 • 6H 2 0) are greatly expanded; 
kainite disappears altogether; epsomite (MgS04 • 7H2O) is slightly reduced, 
and so on (Figure 4.34b). 

When crystallizing from multicomponent systems, kinetic factors often over- 
ride thermodynamic considerations (the so-called Ostwald rule of stages - 
section 5.7). The phase which crystallizes is not necessarily the one which is 
thermodynamically most stable, but the one which crystallizes the fastest. 
Numerous examples of this sort of behaviour are available. 

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5 Nucleation 

The condition of supersaturation or supercooling alone is not sufficient cause 
for a system to begin to crystallize. Before crystals can develop there must exist 
in the solution a number of minute solid bodies, embryos, nuclei or seeds, that 
act as centres of crystallization. Nucleation may occur spontaneously or it may 
be induced artificially. It is not always possible, however, to decide whether 
a system has nucleated of its own accord or whether it has done so under the 
influence of some external stimulus. 

Nucleation can often be induced by agitation, mechanical shock, friction and 
extreme pressures within solutions and melts, as shown by the early experi- 
ments of Young (1911) and Berkeley (1912). The erratic effects of external 
influences such as electric and magnetic fields, spark discharges, ultra-violet 
light, X-rays, 7-rays, sonic and ultrasonic irradiation have also been studied 
over many years (Khamskii, 1969) but so far none of these methods has found 
any significant application in large-scale crystallization practice. 

Cavitation in an under-cooled liquid can cause nucleation, and this probably 
accounts for a number of the above reported effects. Hunt and Jackson (1966) 
demonstrated, by a novel experimental technique, that nucleation occurs when 
a cavity collapses rather than when it expands. Very high pressures (~10 5 bar) 
can be generated by the collapse of a cavity; the change in pressure lowers the 
crystallization temperature of the liquid and nucleation results. It is even 
suggested that nucleation caused by scratching the side of the containing vessel 
could be the result of cavitation effects. 

At the present time there is no general agreement on nucleation nomenclat- 
ure so to avoid confusion the terminology to be used in this and subsequent 
chapters will be defined here. The term 'primary' will be reserved for all cases of 
nucleation in systems that do not contain crystalline matter. On the other hand, 
nuclei are often generated in the vicinity of crystals present in a supersaturated 
system; this behaviour will be referred to as 'secondary' nucleation. Thus we 
may consider a simple scheme: 


(induced by crystals) 


(spontaneous) (induced by foreign particles) 

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182 Crystallization 

5.1 Primary nucleation 

5.1.1 Homogeneous nucleation 

Exactly how a stable crystal nucleus is formed within a homogeneous fluid is 
not known with any degree of certainty. To take a simple example, the con- 
densation of a supersaturated vapour to the liquid phase is only possible after 
the appearance of microscopic droplets, called condensation nuclei, on the 
condensing surface. However, as the vapour pressure at the surface of these 
minute droplets is exceedingly high, they evaporate rapidly even though the 
surrounding vapour is supersaturated. New nuclei form while old ones evap- 
orate, until eventually stable droplets are formed either by coagulation or under 
conditions of very high vapour supersaturation. 

The formation of crystal nuclei is an even more difficult process to envisage. 
Not only have the constituent molecules to coagulate, resisting the tendency to 
redissolve (section 3.7), but they also have to become orientated into a fixed 
lattice. The number of molecules in a stable crystal nucleus can vary from about 
ten to several thousand: water (ice) nuclei, for instance, may contain about 100 
molecules. However, a stable nucleus could hardly result from the simultaneous 
collision of the required number of molecules since this would constitute an 
extremely rare event. More likely, it could arise from a sequence of bimolecular 
additions according to the scheme: 

A +A^A 2 
A2 + A ^ A3 
A n -\ + A ^ A n (critical cluster) 

Further molecular additions to the critical cluster would result in nucleation 
and subsequent growth of the nucleus. Similarly, ions or molecules in a solution 
can interact to form short-lived clusters. Short chains may be formed initially, 
or flat monolayers, and eventually a crystalline lattice structure is built up. The 
construction process, which occurs very rapidly, can only continue in local 
regions of very high supersaturation, and many of the embryos or 'sub-nuclei' 
fail to achieve maturity; they simply redissolve because they are extremely 
unstable. If, however, the nucleus grows beyond a certain critical size, as 
explained below, it becomes stable under the average conditions of super- 
saturation obtaining in the bulk of the fluid. 

The structure of the assembly of molecules or ions which we call a critical 
nucleus is not known, and it is too small to observe directly. It could be 
a miniature crystal, nearly perfect in form. On the other hand, it could be a 
rather diffuse body with molecules or solvated ions in a state not too different 
from that in the bulk fluid, with no clearly defined surface. The morphology 
of very small atomic clusters has been discussed by Hoare and Mclnnes 

The classical theory of nucleation, stemming from the work of Gibbs (1948), 
Volmer (1939), Becker and Doring (1935) and others, is based on the con- 
densation of a vapour to a liquid, and this treatment may be extended to 

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crystallization from melts and solutions. The free energy changes associated 
with the process of homogeneous nucleation may be considered as follows. 

The overall excess free energy, AG, between a small solid particle of solute 
(assumed here, for simplicity, to be a sphere of radius r) and the solute in 
solution is equal to the sum of the surface excess free energy, AGs, i.e. the 
excess free energy between the surface of the particle and the bulk of the 
particle, and the volume excess free energy, AGV, i-e- the excess free energy 
between a very large particle (r = oo) and the solute in solution. AGs is 
a positive quantity, the magnitude of which is proportional to r 2 . In a super- 
saturated solution Gy is a negative quantity proportional to r 3 . Thus 




A-kt 7 - 

3 AG V 


where AG V is the free energy change of the transformation per unit volume and 
7 is the interfacial tension, i.e., between the developing crystalline surface and 
the supersaturated solution in which it is located. The term 'surface energy' is 
often used as an alternative to interfacial tension, but the latter term will be 
used throughout here for consistency. The two terms on the right-hand side of 
equation 5.1 are of opposite sign and depend differently on r, so the free energy 
of formation, AG, passes through a maximum (see Figure 5.1). This maximum 
value, AG cr i t , corresponds to the critical nucleus, r c , and for a spherical cluster 
is obtained by maximizing equation 5.1, setting dAG/dr = 0: 



87IT7 + 4nr 2 AG v = 







Figure 5. 

Size of nucleus, r 
1. Free energy diagram for nucleation explaining the existence of a 'critical nucleus 1 

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184 Crystallization 

27 (5.3) 

where AG,, is a negative quantity. From equations 5.1 and 5.3 we get 

AG cnt = 16 ^ ^ (5 . 4 ) 

3(AG„) 2 3 

The behaviour of a newly created crystalline lattice structure in a supersat- 
urated solution depends on its size; it can either grow or redissolve, but the 
process which it undergoes should result in the decrease in the free energy of the 
particle. The critical size r c , therefore, represents the minimum size of a stable 
nucleus. Particles smaller than r c will dissolve, or evaporate if the particle is a 
liquid in a supersaturated vapour, because only in this way can the particle 
achieve a reduction in its free energy. Similarly, particles larger than r c will 
continue to grow. 

Although it can be seen from the free energy diagram why a particle of size 
greater than the critical size is stable, it does not explain the amount of energy, 
AG CT ii, necessary to form a stable nucleus is produced. This may be explained as 
follows. The energy of a fluid system at constant temperature and pressure is 
constant, but this does not mean that the energy level is the same in all parts of 
the fluid. There will be fluctuations in the energy about the constant mean value, 
i.e. there will be a statistical distribution of energy, or molecular velocity, in the 
molecules constituting the system, and in those supersaturated regions where 
the energy level rises temporarily to a high value nucleation will be favoured. 

The rate of nucleation, /, e.g. the number of nuclei formed per unit time per 
unit volume, can be expressed in the form of the Arrhenius reaction velocity 
equation commonly used for the rate of a thermally activated process: 

J = Aexp(-AG/kT) (5.5) 

where k is the Boltzmann constant, the gas constant per molecule (1.3805 x 
KT^JKT 1 =R/N, where R is the gas constant = 8.314 JK ' mor 1 and 
N = the Avogadro number = 6.023 x 10 23 mof 1 ). 

The basic Gibbs-Thomson relationship (section 3.7) for a non-electrolyte 
may be written 

ln5 = ^ (5.6) 

where S is defined by equation 3.68 and v is the molecular volume; this gives 

-A Gv = ^ = ™ (5.7) 

r v 

Hence, from equation 5.4 

AG cnl = 16 " 7V . (5.8) 

3(krinS) 2 

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\| Experimental 


/ V 


j \ 



j \ 

y \ 

*■ ^ .>»... 

Figure 5.2. Effect of super saturation on the nucleation rate 

and from equation 5.5 
J = A exp 

1 67T7 3 v 2 
3k 3 r 3 (ln5) 2 


This equation indicates that three main variables govern the rate of nucleation: 
temperature, T; degree of supersaturation, S; and interfacial tension, 7. 

A plot of equation 5.9, as shown by the solid curve in Figure 5.2, indicates the 
extremely rapid increase in the rate of nucleation once some critical level of 
supersaturation is exceeded. 

The dominant effect of the degree of supersaturation on the time required for 
the spontaneous appearance of nuclei in supercooled water vapour was calcu- 
lated by Volmer (1925) as 







10 62 years 


10 3 years 




10~ 13 s 

In this case, a 'critical' supersaturation could be said to exist in the region of 
S ~ 4.0, but it is also clear that nucleation would have occurred at any value of 
S > 1 if sufficient time had been allowed to elapse. 
Equation 5.9 may be rearranged to give 

In S 

I67T7 v 


3k 3 r 3 ln(y4//). 



and if, arbitrarily, the critical supersaturation, S C nt, is chosen to correspond to 
a nucleation rate, </, of say, one nucleus per second per unit volume, then 
equation 5.10 becomes 


3 2 

3k 3 T 3 In ,4 



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186 Crystallization 

From equations 5.3 and 5.7 the radius of a spherical critical nucleus at a 
given supersaturation can be expressed as 

r c = 2~/v/kT\nS (5.11) 

For the case of non-spherical nuclei, the geometrical factor 16tt/3 in equations 
5.4 and 5.8-5. 10a must be replaced by an appropriate value (e.g. 32 for a cube). 

Similar expressions to the above may be derived for homogeneous nucleation 
from the melt in terms of supercooling. The volume free energy AG V is given 

AH t AT 

AG y = y* (5.12) 

where T* is the solid-liquid equilibrium temperature expressed in kelvins, 
AT = T* — T is the supercooling and A//f is the latent heat of fusion. The 
radius of a critical nucleus is given by 

2 7 r* 

and the rate of nucleation, from equation 5.9, may be expressed by 

16vT7 3 

J = A exp 

3kT*AHJT t (AT r ) 2 



where T t is the reduced temperature defined by T r = T/T* and AT r = AT/T*, 
i.e. AT r = 1 — T T . Equation 5.14, like equation 5.9, indicates the dominant 
effect of supercooling on the nucleation rate. 

For a wide range of substances, including organic melts, the critical homo- 
geneous nucleation temperature expressed in kelvins is approximately 0.8-0.85 
T*, although for hydrocarbons >Ci5 it may approach 0.95T*. 

The size of the critical nucleus is dependent on temperature, since the volume 
free energy, AG V , is a function of the supercooling, AT, (equations 5.12 and 
5.13) giving 

rcOc(Ar)- 1 (5.15) 

and from equation 5.4 

AG crk cx(AT)- 2 (5.16) 

These relationships are shown in Figure 5.3, where it can be seen that the size of 
a critical nucleus increases with temperature. 

Melts frequently demonstrate abnormal nucleation characteristics, as noted 
in the early work of Tamman (1925). The rate of nucleation usually follows an 
exponential curve (solid curve in Figure 5.2) as the supercooling is increased, 
but reaches a maximum and subsequently decreases (broken curve in 
Figure 5.2). Tamman suggested that this behaviour was caused by the sharp 
increase in viscosity with supercooling which restricted molecular movement 
and inhibited the formation of ordered crystal structures. Turnbull and Fisher 
(1949) quantified this behaviour with a modified form of equation 5.9: 

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1 \ 


1 N 


\ 1 


—*^^ 1 


\ 1 



r c, 

Vc 2 





A6(r z ) 

Size of nucleus,/ - 

Figure 5.3. Effect of temperature on the size and free energy of formation of a critical 
nucleus (T\ < T2) 

J = A' exp 

1 67T7 3 v 2 
3k 3 T 3 (ln5) 2 



which includes a 'viscosity' term. When AG', the activation energy for molecu- 
lar motion across the embryo-matrix interface, is exceptionally large (e.g. for 
highly viscous liquids and glasses) the other exponential term is small because 
under these circumstances S is generally very large. AG' then becomes the 
dominant factor in this rate equation and a decrease in nucleation rate is 

The formation of the glassy state is by no means uncommon; Tamman (1925) 
reported that out of some 150 selected organic compounds, all capable of being 
crystallized fairly easily, over 30 per cent yielded the glassy state on cooling 
their melts slowly. 

Although most reported experimental observations of this reversal of the 
nucleation rate have been confined to melts, it is interesting to note that this 
behaviour has also been observed in highly viscous aqueous solutions of citric 
acid (Figure 5.4) (Mullin and Leci, 1969b). 

Excessive supercooling does not aid nucleation. There is an optimum tem- 
perature for nucleation of a given system (see Figures 5.2 and 5.4) and any 
reduction below this value decreases the tendency to nucleate. As indicated by 
the classical relationship (equation 5.9) nucleation can theoretically occur at 
any temperature, provided that the system is supercooled, but under normal 
conditions the temperature range over which massive nucleation occurs may be 
quite restricted. Therefore, if a system has set to a highly viscous or glass-like 
state, further cooling will not cause crystallization. To induce nucleation the 
temperature would have to be increased to a value in the optimum region. 

The nucleation process has been discussed above in terms of the so-called 
classical theories stemming from the thermodynamic approach of Gibbs and 
Volmer, with the modifications of Becker, Doring and later workers. The main 
criticism of these theories is their dependence on the interfacial tension (surface 
energy), 7, e.g. in the Gibbs-Thomson equation, and this term is probably 
meaningless when applied to clusters of near critical nucleus size. 

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iO -10 -20 

Solution temperature, °C 

Figure 5.4. Spontaneous nucleation in supercooled citric acid solutions: A, 4.6 kg of citric 
acid monohydr ate per kg of 'free' water (T* = 62 °C); B, 7.0 kg/kg (T* = <S5°C). (After 
Mullin and Led, 1969b) 

An empirical approach to the nucleation process is described by Nielsen 
(1964), expressing a relationship between the induction period, ?; nc i (the time 
interval between mixing two reacting solutions and the appearance of crystals) 
and the initial concentration, c, of the supersaturated solution: 

he 1 '" 



where k is a constant and p is the number of molecules in a critical nucleus. It 
was suggested that the induction period, which may range from microseconds 
to days depending on the supersaturation, represents the time needed for the 
assembly of a critical nucleus, although this is an over-simplification (see 
section 5.5). 

The so-called classical theories of homogeneous nucleation and the above 
empirical theory all utilize the concept of a clustering mechanism of reacting 
molecules or ions, but they do not agree on the effect of supersaturation on the 
size of a critical nucleus. The former theories indicate that the size is dependent 
on the supersaturation, whereas the latter theory indicates a smaller but 
constant nucleus size. So far these differences have not been resolved, largely 
owing to the fact that the experimental investigation of true homogenous 
nucleation is fraught with difficulty since the production of an impurity-free 
system is virtually impossible. 

Critical reviews of nucleation mechanisms have been made by, for 
example, Nancollas and Purdie (1964), Nielsen (1964), Walton (1967), 

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Nucleation 189 

Strickland-Constable (1968), Zettlemoyer (1969), Nyvlt et al. (1985) and 
Sohnel and Garside (1992). The recent publication by Kashchiev (2000) is 
noteworthy for its in-depth analyses of the thermodynamics and kinetics of 
homogeneous and heterogeneous nucleation. 

Measurement techniques 

It is only in recent years that suitable techniques have been devised for studying the 
kinetics of homogeneous nucleation. The main difficulties have been the prepara- 
tion of systems free from impurities, which might act as nucleation catalysts, and 
the elimination of the effects of retaining vessel walls which frequently catalyse 

An early attempt to study homogeneous nucleation was made by Vonnegut 
(1948) who dispersed a liquid system into a large number of discrete droplets, 
exceeding the number of heteronuclei present. A significant number of droplets 
were therefore entirely mote-free and could be used for the study of true 
homogeneous nucleation. The dispersed droplet method, however, has many 
attendant experimental difficulties: concentrations and temperatures must be 
measured with some precision for critical supersaturations to be determined; 
the tiny droplets (< 1 mm) must be dispersed into an inert medium, e.g. an oil, 
which will not act as a nucleation catalyst; and any nuclei that form in the 
droplets have to be observed microscopically. 

Variations of the droplet method have since been developed to overcome the 
above difficulties (White and Frost, 1959; Melia and Moffitt, 1964; Komarov, 
Garside and Mullin, 1976), but the reliability of homogeneous nucleation 
studies is still difficult to judge. For example, experimental values of the 
'collision factor' (the pre-exponential factor A in equation 5.9) have frequently 
been reported in the range 10 3 to 10 5 cm -3 s , but as these are well outside the 
range predicted from the Gibbs-Volmer theory (^10 25 ) it is probable that true 
homogeneous nucleation was not being observed in these cases. Another point 
to note is that the interfacial energy term 7, which appears in equation 5.14 to 
the third power, cannot be assumed to be independent of temperature (see 
section 5.6). 

An interesting technique was reported by Garten and Head (1963, 1966) who 
showed that crystalloluminescence occurs during the formation of a three- 
dimensional nucleus in solution, and that each pulse of light emitted lasting 
less than 10~ 7 s corresponds to a single nucleation event. Nucleation rates thus 
measured were close to those predicted from classical theory, with collision 
factors in the range 10 25 to 10 30 cm~ 3 s _1 . In their work on the precipitation of 
sodium chloride in the presence of lead impurities, true homogeneous nuclea- 
tion occurred only at very high supersaturations (S > 14). The nucleation 
process was envisaged as the development in the solution of a molecular cluster, 
as a disordered quasi-liquid, which after attaining critical size suddenly 'clicks' 
into crystalline form. As a result of this high-speed rearrangement, the surface 
of the newly formed crystalline particle may be expected to contain large 
numbers of imperfections that would encourage further rapid crystalline 
growth. As a nucleus appears to be generated in <10~ 7 s, its steady build-up 

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190 Crystallization 

as a crystalline body by diffusion is ruled out (a diffusion coefficient for NaCl 
of 10~ 5 cm 2 s gives a formation time more than ten times greater than the 
pulse period). These observations, therefore, may be taken as strong evidence 
for the existence and development of molecular clusters in supersaturated 

From their work with sodium chloride Garten and Head suggested that 
a critical nucleus can be as small as about 10 molecules. A different order of 
magnitude was proposed by Otpushchennikov (1962), who estimated the sizes 
of critical nuclei by observing the behaviour of ultrasonic waves in melts kept 
just above their freezing point. For phenol, naphthalene and azobenzene, for 
example, he suggested that fewer than 1000 molecules constitute a stable 
nucleus. In contrast to this, the work of Adamski (1963) with relatively insol- 
uble barium salts led to the conclusion that a critical nucleus was about 10~ 15 g, 
and as small as this mass may appear it still represents several million mole- 
cules. It is obvious, therefore, that there are still some widely diverging views on 
the question of the size of a critical nucleus, but this is not surprising as the 
critical size is supersaturation-dependent (equation 5.11) and no consideration 
is given to this important variable by any of the above authors. 

Agitation is frequently used to induce crystallization. Stirred water, for 
example, will allow only about j°C of supercooling before spontaneous nucle- 
ation occurs, whereas undisturbed water will allow over 5 °C. Actually, very 
pure water, free from all extraneous matter, has been supercooled some 40 °C. 
Most agitated solutions nucleate spontaneously at lower degrees of supercool- 
ing than quiescent ones. In other words, the supersolubility curve {Figure 3.9) 
tends to approach the solubility curve more closely in agitated solutions, i.e. the 
width of the metastable zone is reduced. 

However, the influence of agitation on the nucleation process is probably 
very complex. It is generally agreed that mechanical disturbances can enhance 
nucleation, but it has been shown by Mullin and Raven (1961, 1962) that an 
increase in the intensity of agitation does not always lead to an increase in 
nucleation. In other words, gentle agitation causes nucleation in solutions that 
are otherwise stable, and vigorous agitation considerably enhances nucleation, 
but the transition between the two conditions may not be continuous; a portion 
of the curve (see Figure 5.5) may have a reverse slope indicating a region where 
an increase in agitation actually reduces the tendency to nucleate. This phe- 
nomenon, observed with aqueous solutions of ammonium dihydrogen phos- 
phate, magnesium sulphate and sodium nitrate, might be explained by 
assuming that agitation effects can lead to the disruption of sub-nuclei or 
molecular clusters in the solution (section 3.13). 

There has long been an interest in the potential effects on the nucleation 
process of externally applied electrostatic or magnetic fields. There is evidence 
that both homogeneous nucleation and the duration of the nucleation induc- 
tion period (section 5.5) can be influenced. However, the relevance of experi- 
mental data, obtained from small-scale investigations under controlled 
laboratory conditions, to bulk solutions in flow or agitated conditions normally 
encountered in industrial practice (section 9.5) is still the subject of considerable 
controversy (Sohnel and Mullin, 1988c). A detailed account of recent theor- 

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Figure 5.5. Influence of agitation on nucleation, showing a region where increased 
agitation can reduce the tendency to nucleate. (After Mullin and Raven, 1962) 

etical studies on the effect of electric fields on nucleation has been given by 
Kashchiev (2000). 

Spinodal decomposition 

The existence of concentration fluctuations in a multicomponent fluid system is 
an implicit assumption in the Gibbs theory of homogeneous nucleation. Two 
types of phase transition (nucleation) have been postulated, viz. composition 
fluctuations large in degree and infinitesimal in spatial extent (e.g. an infinites- 
imal droplet with properties approaching those of the bulk supercooled phase) 
or infinitesimal in degree and large in extent (e.g. continuous changes of phase). 
Classical nucleation theory, based on the former postulate, requires the further 
assumption that a sharp interface exists between the nucleating (stable) and 
supercooled (unstable) phases. The latter mode of transition, known as spinodal 
decomposition, does not require this assumption; a diffuse interface may be 
considered to exist between the phases. 

The underlying theory for spinodal decomposition rests on Gibbs' derivation 
for the limit of stability of a fluid phase with respect to continuous changes of 
phase, represented by 

d 2 G 



where G is the Gibbs free energy per mole of solution and c is the solution 
concentration. On a phase diagram the locus of such points, representing the 
limit of stability, is referred to as the spinodal (see Figure 5.6). Thus, for 
spinodal decomposition to occur, a spontaneous phase transition is necessary 
and the condition 

(d 2 G/dc 2 ) < 


should apply. Within the spinodal region any phase separation can lower 
the free energy of the system and no nucleation step is required. Outside 

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Figure 5.6. (a) Free energy-composition-temperature surface, showing the location of the 
spinodal; (h) temperature-composition graph of the spinodal 

this boundary, nucleation is essential to effect a phase change. The spinodal 
curve represents the limit of the metastable zone (sections 3.12 and 5.3) 
and is characterized by the condition of zero diffusivity (Myerson and Senol, 

5.1.2 Heterogeneous nucleation 

The rate of nucleation of a solution or melt can be affected considerably by the 
presence of mere traces of impurities in the system. However, an impurity that 
acts as a nucleation inhibitor in one case may not necessarily be effective in 
another; indeed it may even act as an accelerator. No general rule applies and 
each case must be considered separately. 

Many reported cases of spontaneous (homogeneous) nucleation are found on 
careful examination to have been induced in some way. Indeed, it is generally 
accepted that true homogeneous nucleation is not a common event. For 
example, a supercooled system can be seeded unknowingly by the presence of 
atmospheric dust which may contain 'active' particles (heteronuclei). Aqueous 
solutions as normally prepared in the laboratory may contain >10 6 solid 
particles per cm 3 of sizes <1 urn. It is virtually impossible to achieve a solution 
completely free of foreign bodies, although careful filtration can reduce the 
numbers to <10 3 cm~ 3 and may render the solution more or less immune to 
spontaneous nucleation. 

Cases are often reported of large volumes of a given system nucleating 
spontaneously at smaller degrees of supercooling than small volumes. A plaus- 
ible explanation is that the larger samples stand a greater chance of being 
contaminated with active heteronuclei. The size of the solid foreign bodies is 
important and there is evidence to suggest that the most active heteronuclei in 
liquid solutions lie in the range 0.1 to 1 um. 

Heteronuclei play an important role in atmospheric water condensation or 
ice formation (Mason, 1957). Atmospheric nuclei have been classified as 'giant' 
(10 to 1 (im) which remain airborne for limited periods only, 'large' (1 to 

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Nucleation 193 

0.2 um) and 'Aitken' (0.2 to 0.005 um). Particles smaller than about 10~ 3 um are 
not normally found in air because they readily aggregate. Aitken nuclei are so 
called because they are active at the supersaturations produced in an Aitken 
counter, an apparatus in which a known volume of air is rapidly expanded; 
water droplets, formed on the particles, settle and are counted microscopically. 
Aitken nuclei, which occur ~10 4 to 10 5 cm~ 3 in the atmosphere, result from 
industrial smokes and vapours, ocean salts arising from spindrift, land dusts, 
particles from volcanic eruptions and even from outer space (Faraday Discus- 
sions, 1998). 

As the presence of a suitable foreign body or 'sympathetic' surface can 
induce nucleation at degrees of supercooling lower than those required for 
spontaneous nucleation, the overall free energy change associated with the 
formation of a critical nucleus under heterogeneous conditions AG' ait , must 
be less than the corresponding free energy change, AG cr ii, associated with 
homogeneous nucleation, i.e. 

AG' cnl = ^AG crit (5.21) 

where the factor </> is less than unity. 

It has been indicated above, e.g. equation 5.9, that the interfacial tension, 7, 
is one of the important factors controlling the nucleation process. Figure 5.7 
shows an interfacial energy diagram for three phases in contact; in this case, 
however, the three phases are not the more familiar solid, liquid and gas, but 
two solids and a liquid. The three interfacial tensions are denoted by 7 c i 
(between the solid crystalline phase, c, and the liquid 1), 7^ (between another 
foreign solid surface, s, and the liquid) and 7 CS (between the solid crystalline 
phase and the foreign solid surface). Resolving these forces in a horizontal 

7si = 7cs + 7ci cos 6 (5.22) 


cos6> = 7sl ~ 7cs (5.23) 


The angle 8, the angle of contact between the crystalline deposit and the foreign 
solid surface, corresponds to the angle of wetting in liquid-solid systems. 

Liquid (0 

deposit (c) 

7c»— L^ — \ ,„ Yv. 

Solid surface (s) 

Figure 5.7. Interfacial tensions at the boundaries between three phases (two solids, one 

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The factor cf> in equation 5.21 can be expressed (Volmer, 1939) as 

(2 + cos0)(l -cos( 
* = 4 

Thus, when 9 = 180°,cos6> = 
AG' cnl = AG cnl 

T and 0=1, equation 5.21 becomes 

When 9 lies between and 180°, < 1; therefore 

AG' , < AG, 

When 9 = 0, 
AG' , = 

0, and 





The three cases represented by equations 5.25-5.27 can be interpreted as 
follows. For the case of complete non-affinity between the crystalline solid and 
the foreign solid surface (corresponding to that of complete non-wetting in 
liquid-solid systems), 6 = 180°, and equation 5.25 applies, i.e. the overall free 
energy of nucleation is the same as that required for homogeneous or spontan- 
eous nucleation. For the case of partial affinity (cf. the partial wetting of a solid 
with a liquid), < 9 < 180°, and equation 5.26 applies, which indicates that 
nucleation is easier to achieve because the overall excess free energy required is 
less than that for homogeneous nucleation. For the case of complete affinity 
(cf. complete wetting) 8 = 0, and the free energy of nucleation of zero. This case 
corresponds to the seeding of a supersaturated solution with crystals of the 
required crystalline product, i.e. no nuclei have to be formed in the solution. 
Figure 5.8 indicates the relationship between and 9. 

As mentioned above, the heterogeneous nucleation of a solution can occur 
by seeding from embryos retained in cavities, e.g. in foreign bodies or the walls 



5 - 


Contact angle, 8 

Figure 5.8. Ratio of free energies of homogeneous and heterogeneous nucleation as a 
function of the contact angle 

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Nucleation 195 

of the retaining vessel, under conditions in which the embryos would normally 
be unstable on a flat surface. This problem has been analysed by Turnbull 
(1950) for different types of cavity. The maximum diameter of a cylindrical 
cavity which will retain a stable embryo is given by 

*« = i ^ (5-28) 

where AG,, is the volume free energy for the phase transformation. If the system 
is heated, this reducing the supersaturation or supercooling and eliminating all 
embryos in cavities larger than d max , and subsequently cooled, the embryos 
retained in the cavities smaller than d max will grow to the mouth of the cavity. 
They will then act as nuclei only if the cavity size d max > 2r c , where r c is the size 
of a critical nucleus (equation 5.3 or 5.13). 

5.2 Secondary nucleation 

A supersaturated solution nucleates much more readily, i.e. at a lower super- 
saturation, when crystals of the solute are already present or deliberately 
added. The term secondary nucleation will be used here for this particular 
pattern of behaviour to distinguish it from so-called primary nucleation (no 
crystals initially present) discussed in section 5.1. There have been several 
comprehensive reviews of the literature on secondary nucleation (Strickland- 
Constable, 1968; Botsaris, 1976; de Jong, 1979; Garside and Davey, 1980; 
Garside, 1985; Nyvlt et al., 1985). 

Among the early papers on this subject may be mentioned the work of Ting 
and McCabe (1934) who demonstrated that solutions of magnesium sulphate 
nucleated in a more reproducible manner at moderate supersaturations in the 
presence of seed crystals. Similar observations were made in studies with copper 
sulphate (McCabe and Stevens, 1951). 

A particular type of secondary nucleation in KBr solutions was interpreted 
by Gyulai (1948) as evidence for a 'transitional boundary layer' of partially 
integrated units which could be stripped off the crystal surfaces by fluid 
motion. This behaviour was demonstrated by Powers (1963), in a series of 
simple experiments, showing that the movement of a sucrose crystal in a 
supersaturated solution, or the movement of the solution past a stationary 
crystal, produced nuclei. Inert replicas of the crystals did not produce nuclei 
under the same conditions. These results tended to suggest that a fluid mechan- 
ical shearing of weak outgrowths or loosely bonded units from the crystal- 
solution interface was responsible. Sung, Estrin and Youngquist (1973) have 
also invoked the concept of fluid shear in an agitated vessel as a mechanism for 
generating embryos (sub-nuclei) which develop into stable nuclei when swept 
into regions of high supersaturation. 

Strickland-Constable (1968) described several possible mechanisms of sec- 
ondary nucleation, such as 'initial' breeding (crystalline dust swept off a newly 
introduced seed crystal), 'needle' breeding (the detachment of weak out- 
growths), 'polycrystalline' breeding (the fragmentation of a weak polycrystalline 

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196 Crystallization 

mass) and 'collision' breeding (a complex process resulting from the interaction 
of crystals with one another or with parts of the crystallization vessel). 

5.2.1 Contact nucleation 

Clontz and McCabe (1971) showed that at moderate levels of supersaturation, 
crystal contacts readily caused secondary nucleation of MgS04 • 7H2O, but 
crystal-crystal contacts gave up to five times as many nuclei as did crystal- 
metal rod contacts. Furthermore, the faster growing faces produced fewer 
nuclei than did the slower growing faces (Johnson, Rousseau and McCabe, 
1972) indicating a connection between secondary nucleation and the crystal 
growth process. 

Collisions in a liquid medium can initiate complex behaviours. Fracture may 
occur at the point of contact, but substantial hydrodynamic forces can operate 
over the surfaces in the vicinity of the point of contact, giving rise to plastic and 
elastic deformation in the parent crystal. Due to energy absorption, a small 
fragment broken off a crystal by collision could be in a considerably disordered 
state, with many dislocations and mismatch surfaces: in fact, it may be nearer 
to an amorphous glassy condition than to a crystal (Strickland-Constable, 
1979). It is not surprising, therefore, that these small crystalline fragments often 
grow much more slowly than macrocrystals. Indeed, cases have been recorded 
where they do not grow at all (Bujac, 1976; van't Land and Wienk, 1976). 
Ristic, Sherwood and Shripathi (1991) suggest that the formation of varying 
numbers of dislocations and the development of elastic strain in the new inter- 
face are the two main reasons for growth rate dispersion (section 6.2.7) in 
attrition fragments smaller than about 150 (im. 

Crystal-agitator contacts are prime suspects for causing secondary nucle- 
ation in crystallizers, although only those crystals that manage to penetrate the 
fluid boundary layer around the blade will actually be hit. The probability of 
such an impact is directly proportional to the rotational speed of the agitator 
(Nienow, 1976). The relative hardness of the contacting bodies is also a factor 
to consider: a metal impeller gives a much higher nucleation rate than one 
coated with a soft material such as polyethylene (Shah, McCabe and Rousseau, 
1973; Randolph and Sikdar, 1974; Ness and White, 1976; Toyokura, Yamazoe 
and Mogi, 1976). 

Energy-impact models have been developed from the results of attrition and 
breakage studies in agitated vessels using crystals suspended in inert liquids 
(Fasoli and Conti, 1976; Nienow and Conti, 1978). A generalized model to 
quantify nucleation by mechanical attrition, based on Rittinger's law for the 
energy required for producing new surface and the additivity of two attrition 
processes due to crystal-crystal and crystal-impeller collisions, has been pro- 
posed by Kuboi, Nienow and Conti (1984). 

Several hydrodynamic models of secondary nucleation in agitated crystal- 
lizers were applied to experimental data obtained from a 6-L agitated batch 
crystallizer using potassium sulphate by Shamlou, Jones and Djamarani (1990). 
They concluded that the secondary nuclei were produced by an attrition 
process with a turbulent fluid-induced mechanism with critical eddies in the 

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Nucleation 197 

viscous dissipation subrange of the turbulent energy spectrum. An empirical 
attrition model, which relates crystal attrition to crystal size and hold-up, was 
developed by Jager et al. (1991) from data obtained with a 20-L continuous 
evaporative crystallizer using ammonium sulphate. 

Direct observation of impact-induced microattrition at the surfaces of potash 
alum crystals immersed in supersaturated solution (Garside, Rush and Larson, 
1979) indicated that the majority of the fragments produced were in the 
1-10 urn size range and had a supersaturation-dependent size distribution. 
Impact energy and the frequency of impact also have an important influence 
on the number of crystals resulting from contact secondary nucleation (Larson, 

Crystalline fragments smaller than about 1 (im probably do not survive in an 
agitated crystallizer where fluctuations of both temperature and supersatura- 
tion commonly occur. The so-called 'survival theory' (Garabedian and Strick- 
land-Constable, 1972) is based on the Gibbs-Thomson effect (section 3.7) 
which suggests that microcrystals can dissolve in solutions that are supersatur- 
ated with respect to macrocrystals. 

The production of breakage fragments, i.e. secondary nuclei, may not always 
be a direct result of crystal interactions or collisions. Chernov, Zaitseva and 
Rashkovich (1990) have shown that growing crystals containing dislocations, 
defects or inclusions are prone to secondary nucleation through the develop- 
ment of internal stresses which lead to crack formation and the subsequent 
production of breakage fragments, i.e. secondary nuclei. Crack propagation 
initiated by the adsorption of impurity species at defects on crystal surface was 
earlier suggested by Sarig and Mullin (1980) as a possible explanation of an 
observed phenomenon of crystal breakdown in a gently agitated suspension in 
a just-saturated solution that also contained a trace amount of a substance that 
was known to be an active habit modifier. 

5.2.2 Seeding 

Probably the best method for inducing crystallization is to inoculate or seed 
a supersaturated solution with small particles of the material to be crystallized. 
Deliberate seeding is frequently employed in industrial crystallization to effect 
a control over the product size and size distribution (section 7.5.5). 

Atmospheric dust frequently contains particles of the crystalline product 
itself, especially in industrial plants or in laboratories where quantities of the 
material have been handled. Fortuitous seeding from this source can serve to 
prevent the crystallization of thermodynamic unstable phases, e.g., hydrates or 
polymorphs, that might otherwise appear (Ostwald's rule of stages, section 5.7). 

Seed crystals, however, do not necessarily have to consist of the material 
being crystallized in order to be effective; isomorphous substances will fre- 
quently induce crystallization. For example, phosphates will often nucleate 
solutions of arsenates; sodium tetraborate decahydrate (borax) can nucleate 
sodium sulphate decahydrate; phenol can nucleate w-cresol; and so on. The 
success of silver iodide, as an artificial rain-maker, is generally attributed to the 
striking similarity of the Agl and ice crystal lattices. However, there are many 

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198 Crystallization 

cases where lattice similarity does not exist and undoubtedly other factors have 
to be considered. Micro-organisms like Pseudomonas syringae, for example, 
have been used commercially as ice nucleators in the snow-making process 
(Liao and Ng, 1990). 

In laboratory and large-scale crystallizations the first sign of nucleation often 
appears in one given region of the vessel, usually where there is a local high 
degree of supersaturation, such as near a cooling surface or at the surface of 
the liquid. On the other hand, it is not uncommon to find some particular spot 
on the vessel wall or on the stirrer acting as a crystallization centre. The most 
reasonable explanation of this phenomenon is that minute cracks and 
crevices in the surface retain tiny crystals from a previous batch which seed 
the system when it becomes supercooled. It is possible, of course, for some part 
of a metal or glass surface to be in a condition in which it acts as a catalyst for 

Melia and Moffitt (1964) studied secondary nucleation in aqueous solutions 
of KC1 and reported that the nucleation rate was independent of the number of 
seeds added. At a constant cooling rate a time-lag or induction period (section 
5.5) was recorded before secondary nucleation commenced. Cayey and Estrin 
(1967) also observed an induction period with seeded solutions of MgS04 in a 
2-L agitated crystallizer and reported a strange effect of the quantity of seeds 
added: one seed (~lmm, <2mg) was more effective in inducing nucleation 
than 50 mg, but less effective than 500 mg. This anomaly, however, was not 
pursued. They also reported that a crystal was not capable of giving rise to fresh 
nuclei until it had reached a critical size of around 220 um. Rousseau, Li and 
McCabe (1976) suggested a critical size of about 200 um. Toyokura, Mogi and 
Hirasawa (1977) reported that crystals smaller than about 100 um did not 
produce secondary nuclei in a fluidized bed with solutions of K alum super- 
cooled by 3°C. Using the same system at a lower supercooling (2°C) in an 
agitated vessel, Kubota and Fujiwara (1990) demonstrated that the critical size 
could vary between about 200 and 500 um depending on the agitator speed and 
its material of construction. 

There are several reasons why the seed crystal size may be influential in 
secondary nucleation. For example, large seeds generate more secondary nuclei 
in agitated systems than do small seeds because of their greater contact prob- 
abilities and collision energies. Indeed, very small crystals can follow the 
streamlines within the turbulence eddies in vigorously agitated solutions, 
behaving essentially as if they were suspended in a stagnant fluid, rarely coming 
into contact with the agitator or other crystals. Other factors to consider are 
that crystals smaller than about 10 um probably grow much more slowly than 
do macrocrystals (section 6.2.7) and, as mentioned above, some damaged 
crystal fragments may not be capable of growing at all. 

Secondary nucleation was observed to occur in a series of pulses, mainly 
during the latent period (section 5.4), when citric acid solutions were seeded in 
an agitated vessel (Mullin and Leci, 1972). The secondary nucleation rate 
decreased with an increase in the seed size or in the number of seeds of a given 
size. The latent period was drastically reduced by decreasing the seed size, but 
was relatively unaffected by the number of seeds added. Increased supersatura- 

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Nucleation 199 

tion increased secondary nucleation and decreased the latent period. Increased 
agitation increased the desupersaturation rate to a maximum and decreased the 
latent period to a minimum. No evidence of fluid mechanical shearing was 
found, and a mechanism of secondary nucleation based on molecular cluster 
formation in solution was proposed. 

Belyustin and Rogacheva (1966) studied the nucleation of MgS04 • 7H2O 
which crystallizes in enantiomorphic forms (section 1.9) at room temperature. 
When this salt nucleated spontaneously (unseeded), the product crystals were 
mostly left-handed. When the solution was seeded with right-handed crystals, 
the number of product crystals increased and the percentage of left-handed 
crystals in the total product decreased. Increases in solution velocity and super- 
saturation in the presence of a right-handed seed both led to decreases in the 
percentage of right-handed crystals in the product. They concluded, however, 
that secondary nucleation in these cases was not caused by fragmentation. 
Because filtration of the solution retarded both seeded and unseeded nucleation 
they proposed that foreign particles (heteronuclei) coming into contact with the 
seed crystals became activated and initiated nucleation. 

In a similar study, Denk and Botsaris (1972) studied the seeded nucleation of 
sodium chlorate enantiomorphs in non-agitated solution and attempted to 
distinguish between nuclei originating from either the solution or a fixed single 
crystal suspended in the solution. At high supercoolings (> 12 °C) when primary 
nucleation was considered to be the dominant mode, the crystals that devel- 
oped were found to be roughly 50 : 50 d- and L-forms. At supercoolings 
between 12 and 4°C, however, virtually 100% of the developed crystals were 
of the same form as the suspended seed, indicating that the nuclei were derived 
directly from the parent crystal. At supercoolings below about 4°C the propor- 
tion fell to around 60% (Figure 5.9). 

The use of selective seeding as a method for separating solutes in solutions 
supersaturated with two salts was proposed by Rousseau and O'Dell (1980). 
Supersaturated aqueous solutions of potassium sulphate together with either 
potassium chloride or dichromate were seeded with one of the solutes to cause 
secondary nucleation of that substance. After recovering the developed crystals 
by filtration, the filtrate was seeded with the second solute to complete the 

Unintentional seeding 

The deliberate use of seed crystals is common practice in both research labor- 
atory, e.g., to encourage the crystallization of a 'difficult' substance, and in 
industrial plant to exert control over the crystal size distribution of the final 
product (section 8.4.5). On the other hand, unintentional seeding, also fre- 
quently encountered in both laboratory and industry, is an uncontrolled event 
which can often cause considerable frustration and trouble. 

The technical literature abounds with tales, some dating back over 150 years, 
of problems caused by the perverse behaviour of crystallizing systems (e.g., 
Buckley, 1952; Woodward and McCrone, 1975; Dunitz and Bernstein, 1995). 
Xylitol, for example, first prepared in 1891 was considered to be a liquid until 

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ffi lOOr 













tb a — -— ==*& 






1 ..... 

i.i i i . 

6 8 




Figure 5.9. Seeded nucleation of sodium chlorate enantiomorphs in non-agitated solution. 
{After Denk and Botsaris, 1972) 

1941 when a solid form melting at 61 °C unexpectedly crystallized. Two years 
later, another form melting at 94 °C appeared, after which subsequent attempts 
to prepare the lower melting (less stable) polymorph have been unsuccessful. 
Benzophenone and the sugars melibiose, levulose and turanose are all examples 
of former liquids that are now regularly produced in crystalline form. Single 
piezoelectric crystals of anhydrous ethylene diamine tartrate were manufac- 
tured on the industrial scale for many years until suddenly at one plant 
a monohydrate nucleated and grew preferentially. Within weeks the affliction 
spread to a second plant many miles away. In another case, ampicillin, a broad- 
spectrum penicillin, could be readily crystallized as either an anhydrate or 
a trihydrate. Several years later a monohydrate made its appearance, since when 
the anhydrate has never been prepared. The secure patenting of pharmaceutical 
products, usually done at a relatively early stage of the laboratory investigations, 
long before industrial production, has become a complex and difficult matter. 

Undoubtedly, many of the above and other examples have been caused by 
unintentional seeding. Reference has already been made in section 5.1.2 to the 
role atmospheric dust can play as a nucleating agent, noting that even foreign 
bodies in the dust can also act as nucleation promoters. Once a certain crystal- 
line form has been prepared in a laboratory or plant, the working atmosphere 
inevitably becomes contaminated with seeds of the particular material. If later 
a fhermodynamically more stable polymorph or hydrate (pseudopolymorph) 
appears, then seeds of this too will enter the atmosphere and play a dominant 
role. However, it is the speed with which another laboratory or plant, often 
some large distance apart, sometimes even in another country, also become 

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Nucleation 201 

contaminated that has led to suggestions of 'world-wide seeding', a phenom- 
enon which cannot be justified. Seeding is essentially a local problem. There are 
innumerable ways in which seeds can be transferred from one location to 
another without assuming that every spot on the earth has become inoculated. 
For example, personnel travel widely and inadvertently carry contaminating 
seeds with them. Samples of the material are frequently passed from one 
location to another, and so on. As Dunitz and Bernstein (1995) say in their 
entertaining and highly informative paper: 'We believe that once a particular 
polymorph has been crystallized it is always possible to obtain it again; it is only 
a matter of finding the right experimental conditions'. 

5.3 Metastable zone widths 

The lack of success of the classical nucleation theories in explaining the beha- 
viour of real systems has led a number of authors to suggest that most primary 
nucleation in industrial crystallizers is heterogeneous rather than homogeneous 
and that empirical relationships such as 

/ = kA^ (5.29) 

are the only ones that can be justified, /is the nucleation rate, k n the nucleation 
rate constant and Ac max the maximum allowable supersaturation (or meta- 
stable zone width). The exponent «, which is frequently referred to as the 
apparent order of nucleation, has no fundamental significance. It does not give 
an indication of the number of elementary species involved in the nucleation 

However, equation 5.29 is not entirely empirical since it can be derived from 
the classical nucleation relationship (equation 5.9) (Nielsen, 1964; Nyvlt, 1968). 
The nucleation rate may be expressed in terms of the rate at which super- 
saturation is created by cooling, viz. 

J = q6 (5.30) 

where 6 = — d9/dt and q is the mass of crystalline substance deposited per unit 
mass of 'free' solvent present when the solution is cooled by 1 °C. q is a function 
of the concentration change and of the crystallizing species. In general, 

«-.£ (5.31, 

where e = R/[l — c(R — l)] 2 . R is the ratio of the molecular weights of hydrate: 
anhydrous salt and c is the solution concentration expressed as mass of anhy- 
drous solute per unit mass of solvent at a given temperature. 

The maximum allowable supersaturation, Ac max , may be expressed in terms 
of the maximum allowable undercooling, A6> max : 

Ac max = NJLjAfcnax (5.32) 

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202 Crystallization 

so equation 5.29 can be rewritten as 

or, taking logarithms, 




log 9 = (n - 1) log ( -^ J - log £ + log k n + n log A0 max (5.34) 

which indicates that the dependence of log 9 on log A0 max is linear with a line of 
slope n. 

Although equation 5.34 can be useful for characterizing the metastability of 
crystallizing systems, as will be described below, it is no longer regarded as a 
reliable indicator of the nucleation kinetics alone. The over-simplification in the 
above analysis is that it assumes that at the moment when nuclei are first 
detected the rate of supersaturation is equal to the rate of nucleation, but the 
true situation is rather more complex. The created supersaturation is dissipated 
in two ways, partly by growth on existing crystalline particles and partly by the 
formation of new nuclei. Further, in the experimental determination of the 
metastable limit, nuclei are not detected at the moment of their creation but at 
some later time when they have grown to visible size (at say about 10 um). In 
other words, the results of such measurements are dependent not only on 
nucleation but also on the subsequent crystal growth process. 

Recognizing this fact, Nyvlt (1983) proposed a refinement of the theoretical 
analysis and concluded that for unseeded solutions the slope of the 

log Aft llax versus log 9 

line is not equal to n but to (3g + 4 + n)/4 where g is the apparent 'order' of the 
growth process (equation 6.18). 

Janse and de Jong (1978) have warned that attempts to evaluate crystal- 
lization kinetics from metastable zone width evidence should be treated with 
caution, while Kubota, Kawakami and Tadaki (1986) have suggested that the 
cooling rate dependence of A6> max can reasonably be explained by a random 
nucleation model. Other detailed analyses of metastable zone width measure- 
ments and their relationship to nucleation and growth kinetics have been made 
by Mullin and Jancic (1979) and Sohnel and Mullin (1988b). 

The simple apparatus shown in Figure 5.10 (Mullin, Chakraborty and 
Mehta, 1970), based on an earlier one devised by Nyvlt (1968), can be used 
to determine equilibrium solubilities (section 3.9) as well as metastable zone 
widths (section 3.12). About 40 mL of nearly saturated solution of known 
concentration is placed in the 50-mL flask and rapidly cooled until nucleation 
commences. The contents of the flask are then slowly heated. The cooling and 
heating sequences may be effected by means of the water jacket, as shown, or 
by an externally operated cold/hot air blower. On approaching the saturation 
temperature the heating rate is reduced to about 0.2°C/min. The temperature 
at which the last crystalline particle disappears is taken as the saturation 
temperature, 9*. 

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Nucleation 203 

Figure 5.10. Apparatus for measuring metastable limits in agitated solutions: A, cooling 
water-bath; B, pump; C, flow meter; D, magnetic stirrer; E, Perspex water jacket; 
F, thermometer 

The nucleation temperature is measured in a similar way. The flask contain- 
ing the solution of known concentration is warmed to about 4 or 5° higher than 
the saturation temperature. A steady rate of cooling is maintained and the 
temperature at which nuclei first appear is recorded. The difference between the 
saturation and nucleation temperatures is the maximum allowable undercool- 
ing, A6> max , corresponding to a particular cooling rate 6. 

Nucleation temperatures in the presence of crystalline materials can be 
determined by a procedure similar to that for the measurement of unseeded 
data by introducing two small crystals (~ 2 mm in size) into the flask when the 
solution has cooled to its predetermined saturation temperature. 

The variation of the maximum allowable undercooling A# max with the cool- 
ing rate 9 for aqueous solutions of ammonium sulphate (Mullin, Chakraborty 
and Mehta, 1970) is shown in Figure 5.11. The lines for seeded and unseeded 
solutions are not parallel; the seeded points lie approximately 1.5-2 °C below 
the unseeded. The slopes of the lines for seeded and unseeded solutions are 
approximately 2.6 and 6.4, respectively, which indicates that the mechanisms of 
primary and secondary nucleation are different. The best straight lines through 
the data yield the relationships 

(9 = (1.38 ± 0.9)A6» : 


seeded (secondary) 


9 = (1.28 ± 0.91) x 1O- Z A0' 

Q 6.43±1.62 

unseeded (primary) 

which give a measure of the scatter of the data. The maximum allowable 
undercoolings for seeded and unseeded solutions are more or less independent 
of the saturation temperature over the range 20-40 °C, but do depend on the 
rate of cooling. At low rates of cooling (^5°C/h) the values are about 1.8 and 
3.8 °C for seeded and unseeded solutions, respectively, of ammonium sulphate 
compared with 3.5 and 5 °C for a cooling rate of 30 °C/h. 

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O 20 








All seeded 


4 6 

8 10 


4 6 2 4 

Maximum allowable undercooling, AS, 

o5-20 ppm Na* *5-20 ppm Ca 2+ DI5 ppm Fe 3 ~ B30 ppm Fe 3+ 
£5 ppm Cr 3+ A 20 ppm Cr 3+ v40 ppm Cr 3+ 

Figure 5.11. Nucleation characteristics of ammonium sulphate aqueous solution: (a) pure 
solutions, seeded and unseeded; (b) effect of impurities in seeded solutions. The broken line 
represents data from (a). (After Mullin, Chakraborty and Mehta, 1970) 

Undercooling data obtained from unseeded solutions have little or no indus- 
trial relevance. In fact it is often impossible to obtain consistent 'unseeded' 
values for many aqueous solutions, e.g. sodium acetate, sodium thiosulphate 
and citric acid. For crystallizer design purposes, the lowest 'seeded' value 
should be taken as the maximum allowable undercooling, and the working 
value of the undercooling should be kept well below this. 

Some typical maximum allowable undercoolings in seeded solutions are 
given in Table 5.1. It should be noted that although the values of A6> max for 
any two substances may be similar, the values of the supersaturation, Ac max 
and S, may be very different. The relationship between the two quantities is 

Table 5.1. Maximum allowable undercooling* , A0 max , for some common aqueous salt 
solutions at 25 °C (measurements made in the presence of crystals under conditions of slow 
cooling (~5°C/h) and moderate agitation) 









NH 4 alum 


MgS0 4 • 7H 2 






NH 4 C1 


NiS0 4 • 7H 2 


NaHP0 4 • 12H 2 






NaBr • 2H 2 


NaN0 3 




(NH 4 ) 2 S0 4 


Na 2 C0 3 • 10H 2 O 


NaN0 2 


KH 2 P0 4 




Na 2 Cr0 4 • 10H 2 O 


Na 2 S0 4 • 10H 2 O 




C11SO4 • 5H 2 




Na 2 S 2 3 • 5H 2 


KN0 2 


FeS0 4 • 7H 2 


Na 2 B 4 O v • 10H 2 O 


K alum 


K 2 S0 4 


"The working value for normal crystallizer operation may be 50% of these values, or lower. The 
relation between A9 mia and Ac max is given by equation 5.32. 

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Nucleation 205 

given by equation 5.32. For example, A# max = 1 °C for both sodium chloride 
and sodium thiosulphate, but the corresponding values of Ac max are 0.25 
and 18 g of crystallizing substance per kg of solution and S ~ 1.01 and 1.4, 

It has long been known that the metastable zone width can be greatly 
affected by the thermal history of the solution. A solution that has been kept 
for an hour or so at a temperature sufficiently higher than the saturation 
temperature will be found to have a wider metastable zone than if it had been 
kept only slightly above the saturation temperature. The higher the preheating 
and the longer the solution is maintained at that temperature, the higher the 
supersaturation at which nucleation commences. Preheating also increases the 
induction period (section 5.5) and decreases the number of crystals formed 
(Sohnel and Garside, 1992). The influence of thermal history has often been 
attributed to the deactivation of heteronuclei in the solution, but an alternative 
view is that preheating changes the solution structure and influences the sub- 
critical cluster sizes (Nyvlt et ah, 1985). 

The experimental measurement of industrially meaningful metastable zone 
widths can be very time consuming. For this reason Mersmann and Bartosch 
(1998) have proposed a theoretical model claimed to be able to predict working 
values for the design of seeded batch crystallizers. A number of basic assump- 
tions are made. First, that the secondary nucleation is not caused by attrition 
between seed crystals, but by surface nucleation on the seeds which develop into 
outgrowths and later detach. This mode of behaviour was first analysed by 
Nielsen (1964) and given the name 'needle breeding' by Strickland-Constable 
(1979). It is further assumed that the development of the outgrowths is con- 
trolled by the integration step (section 6.1.4) and that the shower of detectable 
nuclei that marks the onset of secondary nucleation occurs when the volumetric 
hold-up of crystals in the vessel is between 10~ 4 and 10~ 3 (m 3 crystals/m 3 
suspension) corresponding to a detectable size of ~ 10 urn. 

5.4 Effect of impurities 

The presence of impurities in a system can affect nucleation behaviour very 
considerably. It has long been known, for example, that the presence of small 
amounts of colloidal substances such as gelatin can suppress nucleation in 
aqueous solution, and certain surface-active agents also exert a strong inhibit- 
ing effect. Traces of foreign ions, especially Cr 3+ and Fe 3+ , can have a similar 
action on inorganic salts, as can be seen from the data recorded in Figure 5.11b. 
It would be unwise to attempt a general explanation of the phenomenon of 
nucleation suppression by added impurities with so little quantitative evidence 
yet available, but certain patterns of behaviour are beginning to emerge. For 
example, the higher the charge on the cation the more powerful the inhibiting 
effect, e.g. Cr 3+ > Fe 3+ > Al 3+ > Ni 2+ > Na+. Furthermore there often 
appears to be a 'threshold' concentration of impurity above which the inhibit- 
ing effect may actually diminish (Mullin, Chakraborty and Mehta, 1970). The 
modes of action of high molecular weight substances and cations are probably 

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206 Crystallization 

quite different. The former may have their main action on the heteronuclei, 
rendering them inactive by adsorbing on their surfaces, whereas the latter may 
act as structure-breakers in the solution phase. 

Other suggestions have been made for the action of impurities. For example, 
Botsaris, Denk and Chua (1972) suggested that if the impurity suppresses 
primary nucleation, secondary nucleation can occur if the uptake of impurity 
by the growing crystals is significant; the seed crystal creates an impurity 
concentration gradient about itself; the concentration of impurity near the 
crystal surface becomes lower than that in the bulk solution; and if it is reduced 
low enough, nucleation can occur. Another possibility is that certain impurities 
could enhance secondary nucleation by adsorbing at defects on existing crystal 
surfaces and, by initiating crack propagation, render the crystals prone to 
disintegration (Sarig and Mullin, 1980). Kubota, Ito and Shimizu (1986), on 
the other hand, have interpreted the effects of ionic impurities on contact 
secondary nucleation by a random nucleation model. 

The presence of soluble impurities can also affect the induction period, t m & 
(section 5.5), but it is virtually impossible to predict the effect. Ionic impurities, 
especially Fe 3+ and Cr 3+ , may increase the induction period in aqueous 
solutions of inorganic salts. Some substances, such as sodium carboxymethyl- 
cellulose or polyacrylamide, can also increase t m< \, whereas others may have 
no effect at all. The effects of soluble impurities may be caused by changing 
the equilibrium solubility or the solution structure, by adsorption or chemisorp- 
tion on nuclei or heteronuclei, by chemical reaction or complex formation 
in the solution, and so on. The effects of insoluble impurities are also 

The effects of soluble impurities on crystal growth and crystallization 
processes in general are discussed in more detail in sections 6.2.8 and 6.4, 

5.5 Induction and latent periods 

A period of time usually elapses between the achievement of supersaturation 
and the appearance of crystals. This time lag, generally referred to as an 
'induction period', is considerably influenced by the level of supersaturation, 
state of agitation, presence of impurities, viscosity, etc. 

The existence of an induction period in a supersaturated system is contrary to 
expectations from the classical theory of homogeneous nucleation (section 
5.1.1), which assumes ideal steady-state conditions and predicts immediate 
nucleation once supersaturation is achieved. The induction period may 
therefore be considered as being made up of several parts. For example, a 
certain 'relaxation time', t r , is required for this system to achieve a quasi- 
steady-state distribution of molecular clusters. Time is also required for the 
formation of a stable nucleus, t a , and then for the nucleus to grow to a 
detectable size, t g . So the induction period, ^ n d, may be written. 

find = t T + t n + t g (5.35) 

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Nucleation 207 

It is difficult, if not impossible, to isolate these separate quantities. The 
relaxation time depends to a great extent on the system viscosity and, hence, 
diffusivity. Nielsen (1964) has suggested that t r ~ 10 _17 Z) _1 , where 
D = diffusivity (m 2 s _1 ). In an aqueous solution of an electrolyte, with 
D ~ 10 -9 m 2 s _1 , the relaxation time would be about 10~ 8 s. In highly viscous 
systems, however, values of D can be extremely low and t r accordingly very 
high. Indeed, some systems can set to a glass before nucleation occurs. The 
nucleation time depends on the supersaturation which affects the size of the 
critical nucleus (section 5.1.1), but its estimation is the subject of speculation 
(Sohnel and Mullin, 1988a). The growth time depends on the size at which 
'nuclei' are detectable and the growth rate applicable to this early stage of 
development. This latter quantity is difficult to predict since the rate of growth 
of a nucleus cannot be assumed to have the same order of magnitude as that of 
a macrocrystal: the mechanism and rate may well be quite different (section 

In some systems, particularly at low supersaturation, another time lag may 
be observed. To distinguish it from the induction period, defined above as the 
point at which crystals are first detected in the system, the term 'latent period' 
will be used, and is defined here as the onset of a significant change in the 
system, e.g. the occurrence of massive nucleation or some clear evidence of 
substantial solution desupersaturation. 

Figure 5.12 indicates some of these events diagrammatically on a typical 
desupersaturation curve. Supersaturation is created at zero time (point A) 
and a certain induction time t m< \ elapses before crystals are first detected (B). 
This point, of course, is not the nucleation time t n (B') since critical-sized nuclei 
cannot be detected; they need a certain time (f; n( i — t n ) to grow into crystals of 
detectable size. However, at point B, and often for a considerable time after- 
wards, no significant changes in the solution may be detected until, at point C, 
sometimes referred to as the end of the latent period, t\ p , rapid desupersatura- 
tion occurs (D). Crystal growth predominates during the desupersaturation 
region. Towards the end of the gradual approach to equilibrium, E, which 
may take hours or days, an ageing process may occur (section 7.2.2). At very 
high supersaturations, the induction time and latent period can be extremely 
short and virtually indistinguishable. 

The presence of seed crystals generally reduces the induction period, but does 
not necessarily eliminate it. Even if the system is seeded at time t = 0, 
a measurable induction period tm& may elapse before new crystals are detected. 
By definition, these are 'secondary' nuclei and they may appear in several 
bursts throughout the latent period, making it difficult to attach any real 
significance to the induction time itself. For these reasons it may be preferable 
to record the latent period as the more practical characteristic of the system. 
Factors that can influence the induction and latent periods and the rate of 
desupersaturation are temperature, agitation, heat effects during crystalliza- 
tion, seed size, seed surface area and the presence of impurities. 

Induction periods are often measured visually, but a different result can 
be recorded if new crystalline matter in the system is detected by more sens- 
itive means, e.g. by laser light scattering or electric zone sensing methods 

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i i i 


Figure 5.12. A desupersaturation curve (diagramatic): c* = 
t n = nucleation time, t m & = induction time, tip = latent period 

equilibrium saturation, 

(section 2.14.2). This variability serves to emphasize the fact that an experi- 
mentally determined t- m & is not, by itself, a fundamental characteristic of a 
crystallizing system. 

In practice, the determination of t m d by conventional methods presents few 
problems so long as it exceeds about 10 s. For example, reacting solutions may 
be quickly mixed in an agitated vessel and the time recorded when the first 
physical property change or the first crystals are detected (Mullin and Osman, 
1973). Serious complications can arise, however, when t- m & is less than about 5 s 
because the mixing time in a simple vessel could be comparable with or even 
exceed the measured induction time. For the successful measurement of short 
induction periods, therefore, two things are essential: (1) very rapid mixing and 
(2) a fast sensitive method for the detection of the appropriate system changes. 

A useful technique for the precipitation of relatively insoluble electrolytes is 
the stopped-flow method (Sohnel and Mullin, 1978b). If two stable solutions, 
which react to form a supersaturated solution of the reactant, are mixed 
together instantaneously, no detectable changes occur for some time. However, 
as soon as the reactant starts to precipitate the concentration of the electrically 
conductive species begins to decrease and this causes the solution conductivity 
to diminish. The period of conductivity steadiness is inversely proportional to 
the supersaturation, and for highly supersaturated solutions it can be less than 
a millisecond. 

A precipitation cell made of Perspex (overall dimensions 100 x 60 x 40 mm) 
is shown in Figure 5.13. The two reactant solutions (5mL each) are placed in 
the separate 10-mm diameter chambers, A, from where they are displaced by 
the twin piston, B, into the mixing chamber, C. The twin piston is rapidly 
plunged by hand, an operation that takes less than 0.1 s, and a microswitch, 
situated at the lowest position of the piston, is triggered when the piston stops 
at the bottom of the feedstock chambers. Two platinum electrodes, D, are 
located in the 4-mm diameter outlet channel, E, at a distance of 15 mm from 

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Figure 5.13. Precipitation cell. (A) reactant solution chambers; (B) twin pistons; (C) mixing 
chamber; (D) disc electrodes; (E) outlet channel. (After Sohnel and Mullin, 1978b) 

the mixing chamber in such a way that they do not obstruct the liquid flow. The 
detection equipment includes a storage oscilloscope which permits time 
measurements from 10 s to 1 |j,s. A microswitch triggers the oscilloscope sweep. 
The sensitivity of the method may be estimated as follows. Two solutions are 
forced into the mixing chamber where they react to form a supersaturated 
solution of the reactant. The supersaturated solution then travels down the 
outlet channel. At some distance, /, from the mixing chamber, the first detect- 
able change in conductivity occurs. The distance, /, which depends on both the 
liquid velocity and the level of supersaturation achieved, is a constant while 
liquid is flowing in the channel, assuming steady-state conditions. However, 
when the flow stops, i.e. when the mixing process is completed, the 'detection 
boundary' in the liquid phase travels back up the channel, towards the mixing 
chamber, with a velocity ljt m< x and it reaches the measuring point, located at 

a distance, d, from the mixing 
liquid flow where 



chamber, in a time f e xp after the cessation of 


If / 3> d then t exp ~ /; nc j, i.e. the experimentally measured time can be regarded 
as being equivalent to the induction period. The limit of application may be 
estimated to lie at / ~ 3d, where / e xp may still be regarded as approximately 
equal to tmd if experimental errors are taken into account. 

A typical curve recorded on the oscilloscope display is shown in Figure 5.14. 
From point A (where liquid movement had stopped and the oscilloscope sweep 
was initiated by the microswitch) to point B, the solution conductivity does not 
change detectably. Then the conductivity suddenly decreases (B to C) and 

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Time »- 

Figure 5.14. A typical oscilloscope record indicating the induction period AB 

continues to decrease slowly over a long period. The time corresponding to the 
interval AB is taken as the induction period of precipitation. The interval BC, 
the length of which is a function of the initial solution supersaturation, is caused 
by the sudden creation of nuclei and their subsequent growth. The last period, 
beyond point C, reflects the final slow growth of the crystals in a solution with a 
near-depleted supersaturation. 

5.6 Interfacial tension (surface energy) 

As the induction period can be affected profoundly by so many external 
influences, it cannot be regarded as a fundamental property of a system. Nor 
can it be relied upon to yield basic information on the process of nucleation. 
Nevertheless, despite its complexity and uncertain composition, the induction 
period has frequently been used as a measure of the nucleation event, making 
the simplifying assumption that it can be considered to be inversely propor- 
tional to the rate of nucleation: 

find oc J 


The classical nucleation relationship (equation 5.9) may therefore be written 


log fi nd OC 


T\\o g sy 

which suggests that, for a given temperature, a plot of log t m & versus ( log S)~ 
should yield a straight line, the slope of which should allow a value of the 
interfacial tension, 7, to be calculated. This can only be justified, however, if the 
data relate to true homogeneous nucleation. In a similar manner, the Arrhenius 
reaction velocity relationship (equation 5.5) written in terms of the induction 



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will allow evaluation of the activation energy of homogeneous nucleation from 
the slope of a linear plot of log tm& versus T~ l . 

An experimentally determined linear relationship between log t m< \ and 
(logS)~ is no guarantee that homogeneous nucleation has occurred, as can 
be seen from stopped-flow precipitation data for CaCCh plotted in Figure 5.14 
(Sohnel and Mullin, 1978b) where two different straight lines can be drawn 
through the experimental points. The change of slope at (logS)~ ~ 0.55, i.e. 
S ~ 20, marks a division between homogeneous and heterogeneous nucleation. 
The slope of the line in the higher supersaturation region to the left of the 
diagram gives a value of 7 ~ 80mJm~ 2 . Data for SrCC>3, an even less soluble 
salt, does not show a transition to heterogeneous nucleation in the supersatura- 
tion range studied (S = 50-70) and a value of 7 ~ 100 mJm~ 2 is calculated 
from the slope of this line. 

Much lower values of 7 are expected for soluble salts. The data in Figure 5.16 
for nickel ammonium sulphate, where t m & was determined visually (Mullin and 
Osman, 1973; Mullin and Ang, 1976), again show a homogeneous/hetero- 
geneous division, this time at a value of S ~ 1.8, and a value of 7 ~ 4mJm -2 
is calculated from data in the left hand region for S > 2. 

The temperature dependence of interfacial tension has been demonstrated 
using induction period data for nickel ammonium sulphate recorded over a 
short temperature range. The salt was precipitated by quickly mixing equimolar 
solutions of nickel and ammonium sulphates after which the system was 
allowed to remain static until nucleation occurred. Plots of log t m d versus 
r~ 3 (logS)~ , in accordance with equation 5.38, gave a family of straight lines 

0.4 0.6 0.8 

Supersaturation, (log S)~ 2 


Figure 5.15. Induction period as a function of initial supersaturation for calcium and 
strontium carbonates. The data for CaCC>3 indicate a transition between homogeneous 
and heterogeneous nucleation. (After Sohnel and Mullin, 1978b) 

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of different slope indicating that 7 increased from 3.9 mJm 2 at 20 °C to 
4.6 mJm- 2 at 0°C (Mullin and Osman, 1973). 

30 60 90 

Supersoturation, (log S)" z 


Figure 5.16. Induction period as a function of initial super saturation for nickel ammonium 
sulphate. A = homogeneous, B = heterogeneous nucleation. (After Mullin and Ang, 1976) 





20 40 60 BO 

T" 3 (l09SI" J xl0 7 

Figure 5.17. Plot of log t versus r~ 3 (log,S)~ for (a) non-agitated and (b) agitated 
systems: (□) 15, (v) 20, (O) 25, and (A) 35 °C. (After Mullin and Zdcek, 1981) 

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Another example is shown in Figure 5.17 for the precipitation of potassium 
alum. The data points for the static system {Figure 5.17a) lie on three straight 
lines giving 7 values, calculated from equation 5.38, ranging from 2.03 mJm -2 
at 35 °C to 3.14mJm~ 2 at 15 °C. A different picture emerges from precipitation 
in agitated solution {Figure 5.17b) when the relationship between logr and 
r~ 3 (logS)~ is non-linear suggesting a heterogeneous or even a secondary 
mode of nucleation (Mullin and Zacek, 1981). A value of 7 around 37mJm~ 2 
has been reported (Lancia, Musmarra and Prisciandaro, 1999) for the sparingly 
soluble CaSC>4 • 2H2O and no significant variation was found over the tem- 
perature range 25-90 °C. 

A graph {Figure 5.18) attempting to relate interfacial tension 7 with equi- 
librium solubility c* was constructed by Nielsen and Sohnel (1971) after asses- 
sing a wide variety of experimental data. The link between 7 and c* can be 
substantiated on the basis of regular solution theory (Bennema and Sohnel, 
1990). Following similar lines, Mersmann (1990) proposed the equation 











.•PbCr0 4 

PbC0 3 

# CaW0 4 
•PbC 2 4 
BaS0 4 

Mg(0H) 2 

m « A g2S04 

BaCr0 4 » \ .CaMo0 4 

BaW0 4 * XboCOj •CHjCOOAq 

. . _ __— -— -^\._--^BaMo0 4 
Ag 2 Cr0 4 Ti 2 Cr0 4 .*Csr M o0 4 

AgCI» PbS0X .TIBr* 0030 -*' 2 ^ 

BaSe0 4 §7s5* X II0 * T ' C ' 

SrC 2 4 -H 2 # 
AgBr« pbse0 4 # TISCN* 

SrW0 4 

:a(0H) 2 

NH 4 Br 

Figure 5. 

-6 -4 -2 2 

Solubility (log C*), mol L" 1 
18. Interfacial tension as a function of solubility. (After Nielsen and Sohnel, 

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214 Crystallization 

derived from fundamental relationships, for predicting interfacial tension 
7(Jm~ 2 ); k= Boltzmann constant (1.38 x 10~ 23 JK~'), N= Avogadro con- 
stant (6.02 x 10 26 kmol _1 ), M = molar mass (kgkmol -1 ), p c = crystal density 
(kgm~ 3 ), cs and Cl are the solute concentrations (kmolm~ 3 ) in the solid and 
liquid phases, respectively. Equation 5.39 appears to be compatible with pub- 
lished data on more than 50 anhydrous salts in aqueous solution. For example, 
for BaSC>4 at 18 °C in saturated aqueous solution T = 291 K, p c = 4500 kg m -3 , 
M = 233kgkmor 1 , c s = 4500/233 = 19.3kmolm- 3 , solubility product K c = 
0.87 x 10- 10 , c L = (K C ) V2 = 0.93 x 10~ 5 kmolm" 3 , and the value of 7 may be 
calculated as approximately 0.12Jm~ 2 . 

A comprehensive review of the general subject of solid material surface 
energy has been made by Linford (1972). 

5.7 Ostwald's rule of stages 

In the early part of the 19th century several workers made the experimental 
observation that some aqueous solutions of inorganic salts, when cooled 
rapidly, first deposited crystals of a less stable form than that which normally 
crystallizes. A frequently quoted example is that of sodium sulphate solution 
which can precipitate heptahydrate crystals at around room temperature before 
the thermodynamically stable decahydrate appears. Another is the crystalliza- 
tion of an unstable polymorph of potassium nitrate in advance of the more 
stable rhombic form. 

Ostwald (1896, 1897) attempted to generalize this sort of behaviour by 
propounding a 'rule of stages' which he stated as: an unstable system does 
not necessarily transform directly into the most stable state, but into one which 
most closely resembles its own, i.e. into another transient state whose formation 
from the original is accompanied by the smallest loss of free energy. Ostwald 
recognized that there were many exceptions to this 'rule' and countless others 
have since been recorded. Thermodynamic explanations alone do not offer 
any theoretical support (Dufor and Defay, 1963; Dunning, 1969), but a com- 
bined thermodynamics-kinetics approach (Cardew and Davey, 1982) does 
appear to offer some justification, although the conclusion is that the rule has 
no general proof. A more recent proposal, based on the assumption of structural 
changes taking place in crystallizing solutions, has been offered as an alternative 
explanation by Nyvlt (1995) together with experimental evidence from aqueous 
solutions of citric acid, ferrous sulphate and sodium hydrogen phosphate. 
Some support for this has been given by a computer simulation of crystal- 
lization from solution (Anwar and Boateng, 1998) which demonstrated the 
development of a diffuse precursor phase, with some elements of crystallinity, 
eventually transforming into a stable crystalline structure. 

Despite the lack of definitive theoretical proof, some form of the rule of 
stages does seem to operate often enough for it to be regarded as important to 
bear in mind when, for example, operating large-scale precipitation processes 
(section 7.2.6). The most probable explanation of the phenomenon lies in the 
kinetics of the transformation, the deciding factor being the relative rates of 

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Nucleation 215 

crystal nucleation and growth of the more-stable and less-stable forms. It is 
in fact, a good example of the behaviour where, if more than one reaction 
is thermodynamically possible, the resulting reaction is not the one that is 
thermodynamically most likely, but the one that has the fastest rate. In other 
words, kinetics are often more important than thermodynamics, and this 
should always be borne in mind when dealing with industrial (non-equilibrium) 
precipitating systems. 

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6 Crystal growth 

6.1 Crystal growth theories 

As soon as stable nuclei, i.e. particles larger than the critical size (section 5.1.1), 
have been formed in a supersaturated or supercooled system, they begin to 
grow into crystals of visible size. The many proposed mechanisms of crystal 
growth may broadly be discussed under a few general headings. 

The surface energy theories are based on the postulation that the shape 
a growing crystal assumes is that which has a minimum surface energy. This 
approach, although not completely abandoned, has largely fallen into disuse. 
The diffusion theories presume that matter is deposited continuously on a 
crystal face at a rate proportional to the difference in concentration between 
the point of deposition and the bulk of the solution. The mathematical analysis 
of the operation is similar to that used for other diffusional and mass transfer 
processes. The suggestion by Volmer (1939) that crystal growth was a discon- 
tinuation process, taking place by adsorption, layer by layer, on the crystal 
surface led to the adsorption-layer theories, several notable modifications of 
which have been proposed in recent years. 

For a comprehensive account of the historical development of the many 
crystal growth theories, reference should be made to the critical reviews by 
Wells (1946), Buckley (1952), Strickland-Constable (1968), Lewis (1980), 
Chernov (1980, 1989) and Nyvlt et al. (1985). 

6.1.1 Surface energy theories 

An isolated droplet of a fluid is most stable when its surface free energy, and 
thus its area, is a minimum. In 1878 Gibbs (1948) suggested that the growth of a 
crystal could be considered as a special case of this principle: the total free 
energy of a crystal in equilibrium with its surroundings at constant temperature 
and pressure would be a minimum for a given volume. If the volume free energy 
per unit volume is assumed to be constant throughout the crystal, then 


minimum (6.1) 

where a, is the area of the rth face of a crystal bounded by n faces, and g/ the 
surface free energy per unit area of the rth face. Therefore, if a crystal is allowed to 
grow in a supersaturated medium, it should develop into an 'equilibrium' shape, 
i.e. the development of the various faces should be in such a manner as to ensure 
that the whole crystal has a minimum total surface free energy for a given volume. 
Of course, a liquid droplet is very different from a crystalline particle; in the 
former the constituent atoms or molecules are randomly dispersed, whereas in 

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Crystal growth 


the latter they are regularly located in a lattice structure. Gibbs was fully aware 
of the limitations of his simple analogy, but in 1885 Curie found it a useful 
starting point for an attempt to evolve a general theory of crystal growth and in 
1901 Wulff showed that the equilibrium shape of a crystal is related to the free 
energies of the faces; he suggested that the crystal faces would grow at rates 
proportional to their respective surface energies. 

The surface energy and the rate of growth of a face, however, should be 
inversely proportional to the reticular or lattice density of the respective lattice 
plane, so that faces having low reticular densities would grow rapidly and 
eventually disappear. In other words, high index faces grow faster than low. 

The velocity of growth of a crystal face is measured by the outward rate of 
movement in a direction perpendicular to that face. In fact to maintain con- 
stant interfacial angles in the crystal (Haiiy's law), the successive displacements 
of a face during growth or dissolution must be parallel to each other. Except for 
the special case of a geometrically regular crystal, the velocity of growth will 
vary from face to face. Figure 6.1a shows the ideal case of a crystal that 
maintains its geometric pattern as it grows. Such a crystal is called 'invariant'. 
The three equal A faces grow at an equal rate; the smaller B faces grow faster; 
while the smallest face C grows fastest of all. A similar, but reverse, behaviour 
may be observed when a crystal of this type dissolves in a solvent; the C face 
dissolves at a faster rate than the other faces, but the sharp outlines of the 
crystal are soon lost once dissolution commences. 

In practice, a crystal does not always maintain geometric similarity during 
growth; the smaller, faster-growing faces are often eliminated, and this mode of 
crystal growth is known as 'overlapping'. Figure 6.1b shows the various stages 
of growth of such a crystal. The smaller B faces, which grow much faster than 
the A faces, gradually disappear from the pattern. 

So far there is no general acceptance of the surface energy theories of crystal 
growth, since there is little quantitative evidence to support them. These the- 
ories, however, still continue to attract attention, but their main defect is their 
failure to explain the well-known effects of supersaturation and solution move- 
ment on the crystal growth rate. 

(a) (b) 

Figure 6.1. Velocities of crystal growth faces: (a) invariant crystal; (b) overlapping 

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218 Crystallization 

6.1.2 Adsorption layer theories 

The concept of a crystal growth mechanism based on the existence of an 
adsorbed layer of solute atoms or molecules on a crystal face was first suggested 
by Volmer (1939). Many other workers have contributed to, and modified 
Volmer's original postulation. The brief account of this subsequent develop- 
ment given below will serve merely to indicate the important features of layer 
growth and the role of crystal imperfections in the growth process. 

Volmer's theory, or as some prefer to call it, the Gibbs-Volmer theory, is 
based on thermodynamic reasoning. When units of the crystallizing substance 
arrive at the crystal face they are not immediately integrated into the lattice, but 
merely lose one degree of freedom and are free to migrate over the crystal face 
(surface diffusion). There will, therefore, be a loosely adsorbed layer of integ- 
rating units at the interface, and a dynamic equilibrium is established between 
this layer and the bulk solution. The adsorption layer, or 'third phase', as it is 
sometimes called, plays an important role in crystal growth and secondary 
nucleation (section 5.3). The thickness of the adsorption layer probably does 
not exceed 10 nm, and may even be nearer 1 nm. 

Atoms, ions or molecules will link into the lattice in positions where the 
attractive forces are greatest, i.e. at the 'active centres', and under ideal condi- 
tions this step-wise build-up will continue until the whole plane face is com- 
pleted (Figure 6.2a and b). Before the crystal face can continue to grow, i.e. 
before a further layer can commence, a 'centre of crystallization' must come 
into existence on the plane surface, and in the Gibbs-Volmer theory it is 

Figure 6.2. A mode of crystal growth without dislocations: (a) migration towards desired 
location; (b) completed layer, (c) surface nucleation 

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Crystal growth 219 

suggested that a monolayer island nucleus, usually called a two-dimensional 
nucleus, is created (Figure 6.2c). 

Expressions for the energy requirement of two-dimensional nucleation and 
the critical size of a two-dimensional nucleus may be derived in a similar 
manner to those for homogeneous three-dimensional nucleation (section 
5.1.1). The overall excess free energy of nucleation may be written 

AG = a 7 + vAG„ (6.2) 

where a and v are the area and volume of the nucleus, and if this is a circular 
disc of radius r and height h, then 

AG = 2nrhj + nr 2 hAG v (6.3) 

and, maximizing to find the critical size, r c , 

— -^ = 2tt/z7 + 2nrhAG r = (6.4) 




AG V , 


In other words, the critical radius of a two-dimensional nucleus is half that of 
a three-dimensional nucleus (equation 5.3) formed under similar environmental 

AG crit = -^- (6.6) 

where AG,, is a negative quantity; so from equation (5.7) 

AG - = krb (6 - 7) 

In a similar manner to that described earlier, the rate of two-dimensional 
nucleation, /', can be expressed in the form of the Arrhenius reaction velocity 

J' = B- exp(- AG cr i t /kT ) (6.8) 


J = B ■ exp 


k 2 r 2 lnS 


Comparing equations 5.8 and 6.7 it can be seen that the ratio of the energy 
requirements of three- to two-dimensional nucleation (sphere : disc) is 
167v/3/ik7 , ln5. By inserting some typical values, e.g. 7=10~ 1 Jm~ 2 , 
v = 2 x l(T 29 m 3 , h = 5 x l(T 10 m, kl = 4x 10 21 J, it can be calculated that 
the ratio is about 50 : 1 for a supersaturation of S = 1.1 and about 1.2 : 1 for 
S = 10. In general, therefore, it may be said that a reasonably high degree of 

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220 Crystallization 

Figure 6.3. Kossel's model of a growing crystal surface showing flat surfaces (A), steps 
(B), kinks (C), surface-adsorbed growth units (D), edge vacancies (E) and surface 
vacancies (F) 

local supersaturation is necessary for two-dimensional nucleation to occur, but 
lower than that required for the formation of three-dimensional nuclei under 
equivalent conditions. 

The Kossel (1934) model of a growing crystal face is depicted in Figure 6.3. It 
envisages that an apparently flat crystal surface is in fact made up of moving 
layers (steps) of monatomic height, which may contain one or more kinks. In 
addition, there will be loosely adsorbed growth units (atoms, molecules or ions) 
on the crystal surface and vacancies in the surfaces and steps. Growth units are 
most easily incorporated into the crystal at a kink; the kink moves along the 
step and the face is eventually completed. A fresh step could be created by 
surface nucleation, and this frequently commences at the corners. 

A crystal should grow fastest when its faces are entirely covered with kinks, 
and the theoretical maximum growth rate can be estimated (equation 6.37). It is 
unlikely, however, that the number of kinks would remain at this high value for 
any length of time; it is well known, for example, that broken crystal surfaces 
rapidly 'heal' and then proceed to grow at a much slower rate. However, many 
crystal faces readily grow at quite fast rates at relatively low supersaturation, 
far below those needed to induce surface nucleation. Crystals of iodine, for 
example, can be grown from the vapour at 1 per cent supersaturation at rates 
some 10 1000 times greater than those predicted by classical theory (Volmer and 
Schultz, 1931)! So it must be concluded that the Kossel model, and its depend- 
ence on surface nucleation, is unreasonable for growth at moderate to low 

A solution to the dilemma came when Frank (1949) postulated that few 
crystals ever grow in the ideal layer-by-layer fashion without some imperfection 
occurring in the pattern. Most crystals contain dislocations (see section 1.13) 
which cause steps to be formed on the faces and promote growth. Of these the 
screw dislocation (section 1.13.2) is considered to be important for crystal 
growth, since it obviates the necessity for surface nucleation. Once a screw 
dislocation has been formed, the crystal face can grow perpetually 'up a spiral 
staircase'. Figure 6.4a-c indicates the successive stages in the development of a 
growth spiral starting from a screw dislocation. The curvature of the spiral 
cannot exceed a certain maximum value, determined by the critical radius for a 
two-dimensional nucleus under the conditions of supersaturation in the med- 
ium in which the crystal is growing. 

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Crystal growth 




(a) (b) (c) 

Figure 6.4. Development of a growth spiral starting from a screw dislocation 

An example of a circular growth spiral on a silicon carbide crystal is shown in 
Figure 6.5. The major and minor axes of the elliptical spirals on the (100) face of 
an ammonium dihydrogen phosphate crystal growing in aqueous solution 
{Figure 6.6) point in the [010] and [001] directions respectively, indicating that 
surface diffusion is faster in the former direction (Davey and Mullin, 1974). The 
polygonized spiral on the C36 hydrocarbon crystal (Figure 6.7) is probably only 
a few long-chain molecules in height. Quite often very complex spirals develop, 
especially when several screw dislocations grow together. Many examples of 
these are shown in the books by Verma (1953) and Read (1953). 

As a completely smooth face never appears under conditions of spiral 
growth, surface nucleation is not necessary and the crystal grows as if the 
surface were covered with kinks. Growth continues uninterrupted at near the 
maximum theoretical rate for the given level of supersaturation. The behaviour 
of a crystal face with many dislocations is practically the same as that of 
a crystal face containing just one. Burton, Cabrera and Frank (1951) developed 

Figure 6.5. A circular spiral on a silicon carbide crystal. (Courtesy of the Westinghouse 

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Figure 6.6. An elliptical spiral on the (100) face of an ammonium dihydrogen phosphate 
crystal growing in aqueous solution (Davey and Mullin, 1974) 

Figure 6.7. A polygonized spiral on the face of a C 36 normal alkane crystal. (Courtesy of 
R. Boistelle) 

a kinetic theory of growth in which the curvature of the spiral near its origin 
was related to the spacing of successive turns and the level of supersaturation. 
By the application of Boltzmann statistics they predicted kink populations, and 
by assuming that surface diffusion is an essential step in the process they were 
able to calculate the growth rate at any supersaturation. 

The Burton-Cabrera-Frank (BCF) relationship may be written 

R = Act 2 tanh(B/cr) 


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Crystal growth 


Supersaturotion, u 

Figure 6.8. The Burton-Cabrera-Frank (BCF) super saturation-growth relationship 
(I, R oc a 2 ; II, an approach to R oc a) 

where R = crystal growth rate. The supersaturation a = S — 1 where S = cjc* 
(see section 3.12.1). A and B are complex temperature-dependent constants 
which include parameters depending on step spacings. 

At low supersaturations the BCF equation approximates to R oc a 2 , but at 
high supersaturations R oc a. In other words, it changes from a parabolic to 
a linear growth law as the supersaturation increases. The volume diffusion 
model proposed by Chernov (1961) gives the same result. The general form of 
these expressions is shown in Figure 6.8. 

It should be pointed out that the BCF theory was derived for crystal growth 
from the vapour; and while it should also apply to growth from solutions (and 
melts), it is difficult to quantify the relationships because of the more complex 
nature of these systems. Viscosities, for example, are higher and diffusivities 
lower in solutions (~10~ 3 Nsm~ 2 (1 cP) and 10 _9 m 2 s _1 ) than in vapours 

(~10~ 5 Nsm~ 2 and 10 _4 m 2 s _1 ). In addition, the dependence of diffusivity 
on solute concentration can be complex (section 2.4). Transport phenomena 
in ionic solutions can be complicated, especially if the different ions exhibit 
complex hydration characteristics. Furthermore, little is known about surface 
diffusion in adsorbed layers, and ion dehydration in or near these layers must 
present additional complicating factors. 

For a comprehensive account of the relationships between the various sur- 
face and bulk diffusion models of crystal growth, and their relevance to crystal 
growth, reference may be made to the reviews by Bennema (1968, 1969, 1984) 
and Chernov (1980, 1989, 1993). 

6.1.3 Kinematic theories 

Two processes are involved in the layer growth of crystals, viz. the generation 
of steps at some source on the crystal face followed by the movement of layers 
across the face. Consideration of the movement of macrosteps of unequal 
distance apart (BCF theory considers a regular distribution of monoatomic 
steps) led Frank (1958) to develop a 'kinematic' theory of crystal growth. The 

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224 Crystallization 

t l 



Figure 6.9. Two-dimensional diagrammatic representation of steps on a crystal face 

step velocity, u, depends on the proximity of the other steps since all steps are 
competing units (Figure 6.9). Thus 

u = qjn (6-11) 

where q is the step flux (the number of steps passing a given point per unit time) 
and n is the step density (the number of steps per unit length in a given region). 
The distance between steps, A = n~ l . The slope of the surface,/?, with reference 
to the close packed surfaces, i.e. the flat ledges, is given by 

p = Vane = hn (6.12) 

and the face growth rate, v, normal to the reference surface by 

v = hq = hnu (6.13) 

where h is the step height. 

If the steps are far apart (9 — > 0), and the diffusion fields do not interfere with 
one another, the velocity of each step, u, will be a maximum. As the step 
spacings decrease and the slope increases, u decreases to a minimum at 
hn= 1 (9 = 45°). Looking at it another way: as the slope increases, the face 
growth velocity v (= u tan 0) increases, approaches a flat maximum and then 
decreases to zero. The shape of this v(p) curve, which is affected by the presence 
of impurities, is an important characteristic of the growth process. 

For the two-dimensional case depicted in Figure 6.9 another velocity, 
c = dx/dt, may be defined which represents the motion of 'kinematic waves' 
(regions on the crystal surface with a constant slope p and velocity v). These 
waves do not contain the same monomolecular steps all the time, as the step 
velocity u = v/p can be greater or less than c. When two kinematic waves of 
different slope meet, a discontinuity in slope occurs, giving rise to 'shock waves' 
across the surface. 

Another problem that can be treated on the basis of the kinematic theory is 
that of step bunching. The steps that flow across a face are usually randomly 
spaced and of different height and velocity. Consequently they pile-up or 
bunch. Growth, and dissolution, can be characterized by the relationship 
between the step flux, q, and step density, n. Two general forms of this relation- 
ship can be considered depending upon whether d q/dn 2 < (Type I) or 
d q/dn 2 > (Type II). The former is analogous to the flow of traffic along a 
straight road and the latter to flood water on a river (see Figure 6.10). 

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Crystal growth 



x> : :1:':*y 




'1 w. 








Figure 6.10. (a) Step flux density curves: type I, d q/dn 2 < 0; type II, d q/dn 2 > 0. 
(b) Surface profiles arising from bunches with type I and type II kinetics, respectively 

6.1.4 Diffusion-reaction theories 

The origin of the diffusion theories dates back to the work of Noyes and 
Whitney (1897) who considered that the deposition of solid on the face of 
a growing crystal was essentially a diffusional process. They also assumed that 
crystallization was the reverse of dissolution, and that the rates of both pro- 
cesses were governed by the difference between concentration at the solid 
surface and in the bulk of the solution. An equation for crystallization was 
proposed in the form 


k m A(c - c*) 


where m = mass of solid deposited in time t; A = surface area of the crystal; 
c = solute concentration in the solution (supersaturated); c* = equilibrium 
saturation concentration; and k m = coefficient of mass transfer. 

On the assumption that there would be a thin stagnant film of liquid adjacent 
to the growing crystal face, through which molecules of the solute would have 
to diffuse, Nernst (1904) modified equation 6.14 to the form 

dm D 


length of the diffusion 

d? S 

where D = coefficient of diffusion of the solute, and S ■ 

The thickness S of the stagnant film would obviously depend on the relative 
solid-liquid velocity, i.e. on the degree of agitation of the system. Film thick- 
nesses up to 150 um have been measured on stationary crystals in stagnant 
aqueous solution, but values rapidly drop to virtually zero in vigorously 
agitated systems. As this could imply an almost infinite rate of growth in 
agitated systems, it is obvious that the concept of film diffusion alone is not 
sufficient to explain the mechanism of crystal growth. Furthermore, crystal- 
lization is not necessarily the reverse of dissolution. A substance generally 
dissolves at a faster rate than it crystallizes at, under the same conditions of 
temperature and concentration. 

Another important finding was made by Miers (1904), who determined, by 
refractive index measurements, the solution concentrations near the faces of 

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crystals of sodium chlorate growing in aqueous solution; he showed that the 
solution in contact with a growing crystal face is not saturated but super- 

In the light of these facts, a considerable modification was made to the 
diffusion theory of crystal growth by Berthoud (1912) and Valeton (1924), 
who suggested that there were two steps in the mass deposition, viz. a diffusion 
process, whereby solute molecules are transported from the bulk of the fluid 
phase to the solid surface, followed by a first-order 'reaction' when the solute 
molecules arrange themselves into the crystal lattice. These two stages, occur- 
ring under the influence of different concentration driving forces, can be 
represented by the equations 


k&A(c — Ci) (diffusion) (6.16) 




k r A(ci — c*) (reaction) 


where /c<j = a coefficient of mass transfer by diffusion; k r = a rate constant for 
the surface reaction (integration) process; and c; = solute concentration in the 
solution at the crystal-solution interface. 

A pictorial representation of these two stages is shown in Figure 6.11 where 
the various concentration driving forces can be seen. It must be clearly under- 
stood, however, that this is only diagrammatic: the driving forces will rarely be 
of equal magnitude, and the concentration drop across the stagnant film is not 
necessarily linear. Furthermore, there appears to be some confusion in recent 
crystallization literature between this hypothetical film and the more funda- 
mental 'boundary layers' (see section 6.3.2). 

Equations 6.16 and 6.17 are not easy to apply in practice because they 
involve interfacial concentrations that are difficult to measure. It is usually 
more convenient to eliminate the term c; by considering an 'overall' concentra- 
tion driving force, c — C* , which is quite easily measured. A general equation for 
crystallization based on this overall driving force can be written as 

Adsorption layer 

Driving force 
for reaction 

Bulk of solution 

N Crystal: solution interface 

Figure 6.11. Concentration driving forces in crystallization from solution according to the 
simple diffusion-reaction model 

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A(c - c*) 


l/ki + l/k r 



Kq~ = 

1 1 
k d k r 

Crystal growth 227 


— =K G A(c-c*y (6.18) 


where Kq is an overall crystal growth coefficient. The exponent g is usually 
referred to as the 'order' of the overall crystal growth process, but the use of 
this term should not be confused with its more conventional use in chemical 
kinetics, where the order always refers to the power to which a concentration 
should be raised to give a factor proportional to the rate of an elementary 
reaction. In crystallization work the exponent, which is applied to a concentra- 
tion difference, has no fundamental significance and cannot give any indication 
of the number of elementary species involved in the growth process. 

If g = 1 and the surface reaction (equation 6.17) is also first-order, the inter- 
facial concentration, c u may be eliminated from equations 6.16 and 6.17 to give 



A. G K-d K-r 


K G = ^ (6.21) 

kd + k r 

For cases of extremely rapid reaction, i.e. large k r , Kq « kd and the crystal- 
lization process is controlled by the diffusional operation. Similarly, if the value 
of kd is large, i.e. if the diffusional resistance is low, Kq m k T , and the process is 
controlled by the surface integration. It is worth pointing out that whatever the 
relative magnitude of kd and k t they will always contribute to Kq. 

The diffusional step (equation 6.16) is generally considered to be linearly 
dependent on the concentration driving force, but the validity of the assumption 
of a first-order surface reaction (equation 6.17) is highly questionable. Many 
inorganic salts crystallizing from aqueous solution give an overall growth rate 
order, g, in the range 1 to 2. The rate equations, therefore, may be written 

1 dm 

R G = = kAc — c;) (diffusion) (6.22) 

A dt 

= k r {c[ — c*Y (reaction) (6.23) 

= K G (c - c*f (overall) (6.24) 

The reverse process of dissolution may be represented by the overall relation- 

^ D = K D (c* - c) d (6.25) 

where d is generally, but not necessarily, unity. From equation 

d = c- Ro/kd (6.26) 

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228 Crystallization 

so equation 6.23 representing the surface integration step, may be written 

R G = k r (\c-^\ (6.27) 

where Ac = c — c* and r > 1 . If r = 1, 

k d k r 


kd + k r 

Ac (6.28) 

as in equation 6.21. However, if r ^ 1, the surface integration step is dependent 
on the concentration driving force in non-linear manner. For example, if 
r = 2, equation 6.27 can be solved to give 

R Q = k A 

2k t Ac 

Ac (6.29) 

However, apart from such simple cases, equation 6.27 cannot be solved 
explicitly for Rq and the relationship between the coefficients Kq, ^d and k r 
remains obscure. Recently, however, Sobczak (1990) has proposed an integral 
method, based on a linearization of equation 6.27, which allows reasonable 
values of fed and k r to be estimated. 

Effectiveness factors 

A quantitative measure of the degree of diffusion or surface integration control 
may be made through the concept of effectiveness factors. A crystal growth rate 
effectiveness factor, r] c , may be defined (Garside, 1971; Garside and Tavare, 
1981) as the ratio of the growth rate at the interface conditions to the growth 
rate expected if the interface were exposed to the bulk solution conditions, or 

rye = (1 - Vc Da) r (6.30) 

where r is the 'order' of the surface integration process, and Da is the 
Damkohler number for crystal growth, which represents the ratio of the 
pseudo-first-order rate coefficient at the bulk conditions to the mass transfer 
coefficient, defined by 

Da = k t (c-c*y-\l-uj)k d l (6.31) 

where uj is the mass fraction of solute in solution. The plot of equation 6.30 in 
Figure 6.12 shows that when Da is large, the growth is diffusion controlled 
(r/ c — > Da~ l ) and when Da is small, the growth is surface integration controlled 

fob - 1). 

Other contributing steps 

It might be thought possible that the diffusional and surface reaction coeffi- 
cients could be quantified by making certain assumptions. For example, if it is 
assumed that the diffusional mass transfer coefficient, kd, in the crystallization 

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Crystal growth 229 

- 0.8 


* 0.4 


n c = 1.0 ( Pure surface integration growth) 

n c ,= Da' 
\ ( Pure diffusion growth) 




Damkbhler number, Da 

Figure 6.12. The effectiveness factor for crystal growth (equation 6.30). (After Garside and 
Tavare, 1981) 

process is the same as that measured for crystal dissolution in near-saturated 
solutions under the same concentration driving force, temperature, etc., then 
values of k T can be predicted. 

Such calculations have been made (Garside and Mullin, 1968; Mullin and 
Gaska, 1969), but the assumption that the diffusion step in crystal growth can 
be related to the diffusion step in dissolution may not always be valid. It is 
possible, for example, that even dissolution is not a simple one-step process. 
Indeed some form of surface reaction (disintegration) step has been measured 
for the dissolution of lead sulphate in water (Bovington and Jones, 1970). 

In any case, the growth process is undoubtedly much more complex than the 
simple two-step process envisaged above. For an electrolyte crystallizing from 
aqueous solution, for example, the following processes may all be taking place 

1 . Bulk diffusion of hydrated ions through the diffusion boundary layer 

2. Bulk diffusion of hydrated ions through the adsorption layer 

3. Surface diffusion of hydrated or dehydrated ions 

4. Partial or total dehydration of ions 

5. Integration of ions into the lattice 

6. Counter-diffusion of released water through the adsorption layer 

7. Counter-diffusion of water through the boundary layer 

The potential importance of the ion dehydration step in the crystallization of 
electrolytes from aqueous solution has been discussed by several authors (Reich 
and Kahlweit, 1968; Nielsen, 1984), and there is evidence that an allowance for 
these effects could account substantially for discrepancies between theoretical 

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230 Crystallization 

and actual growth rates. However, any one of the above processes could 
become rate-controlling, and a rigorous solution of the problem is virtually 
unattainable. Furthermore, the thicknesses of the different layers and films 
cannot be known with any certainty. Adsorbed molecular layers probably do 
not exceed 10~ 2 |J.m; partially disordered solution near the interface may 
account for another 10 -1 |xm; and the diffusion boundary layer is probably 
not much thicker than about 10 urn (see section 6.3.2). 

The individual constants k& and k r are not only difficult if not impossible to 
determine, but can vary from face to face on the same crystal. It is even possible 
for kd to vary over one given face: although it is true that the solution in contact 
with growing crystal face is always supersaturated, the degree of supersaturation 
can vary at different points over the face. In general, the supersaturation is highest 
at the corners and lowest at the centre of the face (Berg, 1938; Bunn, 1949). 

The diffusion theories of crystal growth cannot yet be reconciled with the 
adsorption layer and dislocation theories. It is acknowledged that the diffusion 
theories have grave deficiencies (they cannot explain layer growth or the facet- 
ing of crystals, for example), yet crystal growth rates are conveniently measured 
and reported in diffusional terms. The utilization of the mathematics of mass 
transfer processes makes this the preferred approach, from the chemical engin- 
eer's point of view at any rate, despite its many limitations. 

If a crystallization process were entirely diffusion-controlled or surface reac- 
tion controlled, it should be possible to predict the growth rate by fundamental 
reasoning. In the case of diffusion-controlled growth, for example, the molecu- 
lar flux, F (mols -1 cm -2 ) is related to the concentration gradient, dc/dx, by 

F = D(dc/dx) (6.32) 

where x is the length of the diffusion path and D is the diffusion coefficient. 
Therefore the rate of diffusion, dn/dt (mol s~ ), to a spherical surface, distance 
r from the centre, is given by 

dn i dc 

_=4vTr 2 Z)— (6.33) 

dt dr 

At any instant dnjdt is a constant, so equation 6.33 may be integrated to give 

t Cl dn t n dr 

W„ "si ■? <« 4 > 


dn 4irD(c 2 - Ci) 

d7= i_l (6 ' 35) 

r\ r 2 

If c\ = c* (equilibrium saturation) at r\ = r (the surface of the sphere) and 
c'2 = c (the bulk liquid concentration) at r 2 = oo (i.e. r 2 3> r\), then 

dn . dr 4irr 2 

— =4nrD(c-c* ) = - 6.36 

d; d; v 

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Crystal growth 


So the general equation for the diffusion-controlled linear growth rate may be 
expressed as 


Dv(c - c*) 


The same relationship may be used for the reverse process of dissolution. For 
dissolution into a pure solvent (c = 0) integration of equation 6.37 gives 

r z = Tq - 2Dvc*t 

where ro is the initial size at time t 
(r = 0) is thus given by 

0. The time for complete dissolution 



Substitution of typical values into equation 6.39 leads to some interesting 
observations. Small crystals of reasonably soluble salts may dissolve in frac- 
tions of a second, but those of sparingly soluble substances can take very long 
periods of time. For example, a lum crystal (r = 5 x 10~ 7 m) of lead chromate 
(D« 10" 9 m 2 s-', vK5xl0" 5 m 3 mor', c* « lO^molrrT 3 ) would take 
about 7h to dissolve in water at room temperature. A 10 (im crystal would 
take about 30 days. Tiny crystalline fragments of relatively insoluble substances 
may therefore remain undissolved in unsaturated solutions and act as nuclei in 
subsequent crystallization operations. The behaviour of precipitates attributed 
to the past history of the system may well be associated with this behaviour. 

6.1.5 Birth and spread models 

Several growth models based on crystal surface (two-dimensional) nucleation, 
followed by the spread of the monolayers have been developed in recent years 
(O'Hara and Reid, 1973; van der Eerden, Bennema and Cherepanova, 1978). 
The term 'birth and spread' (B + S) model will be used here, but other names 
such as 'nuclei on nuclei' (NON) and 'polynuclear growth' may also be seen in 
the literature to describe virtually the same behaviour. As depicted in Figure 
6.13, growth develops from surface nucleation that can occur at the edges, 
corners and on the faces of a crystal. Further surface nuclei can develop on the 
monolayer nuclei as they spread across the crystal face. 

Figure 6.13. Development of polynuclear growth by the birth and spread (B + S) 

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232 Crystallization 

The B + S model results in a face growth velocity-supersaturation relation- 
ship of the form 

v = A l0 - 5/6 exp(A 2 /o-) (6.40) 

where A] and A2 are system-related constants. Equation 6.40 is interesting in 
that it describes the only growth model that allows a growth order, g, greater 
than 2. 

6.1.6 Combinations of effects 

Pure diffusion-controlled growth for all sizes in a crystal population is unlikely. 
From diffusional considerations Turnbull (1953) derived the mass flux, N, to 
a growing particle of radius r as 


\dtj D + k y ' 

where D = diffusivity, K is a constant and k is an interface transfer coefficient 
defined by N = n(c r — c*). Concentrations c, c r and c* refer to the bulk solu- 
tion, particle surface and equilibrium saturation, respectively. Integration of 
equation 6.41 gives 


— + - = Ktc (6.42) 

2D k 

For r — > this becomes 

r « kK?c (6.43) 

indicating that the growth of very small nuclei should be interface-controlled. 
For large values of r 

r -> y/(2DKtc) (6.44) 

indicating diffusion control. 

A further complicating factor in using diffusion-controlled growth rate 
expressions such as equation 6.37 is the fact that very small crystals can have 
solubilities significantly higher than those of macrocrystals (Gibbs-Thomson 
effect, section 3.7). In any complete analysis of the growth process, therefore, 
the combined effects of diffusion, surface integration and size-solubility may 
have to be considered together. An analysis along these lines by Matz (1970) 
provided results that appeared to be consistent with experimental data for the 
growth of sodium chloride crystals from aqueous solution. In another 
approach, Leubner (1987) developed a combination model for crystal forma- 
tion by the precipitation of sparingly soluble compounds, e.g. the silver halides, 
which relates the number of stable crystals formed to the precipitation condi- 
tions and to the crystal growth mechanism. 

It is quite possible for more than one basic growth mechanism to influence 
the crystal growth rate simultaneously. When two mechanisms act in parallel, 
e.g. BCF and B + S, the individual rates are additive, and the one that gives 

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Crystal growth 233 

faster rate is rate-determining. When the two mechanisms act consecutively, 
e.g. bulk diffusion followed by BCF growth, they have to share the driving 
force and the slower one (at equal driving force) will be rate determining 
(Nielsen, 1984). 

The combined effects of nucleation and growth on the development of crystal 
populations in crystallizers are discussed in section 6.9.4. 

6.1.7 Crystal surface structure 

The structure of a growing crystal surface at its interface with the growth 
medium, e.g. a supersaturated solution, has an important bearing on the 
particular mode of crystal growth adopted. This property has been character- 
ized by a quantity variously designated as a surface roughness or surface 
entropy factor, or more frequently nowadays simply as the alpha factor (Jack- 
son, 1958; Tempkin, 1964; Bennema and van der Eerden, 1977) which may be 
defined by 

a = £AH/kT (6.45) 

where £ is an anisotropy factor related to the bonding energies in the crystal 
surface layers, AH is the enthalpy of fusion and k is the Boltzmann constant. 
Although reliable a values are not easy to calculate, it is possible, making 
certain simplifying assumptions (Davey, 1982), to make estimates from solu- 
bility data. Values of a < 2 are taken to be indicative of a rough (i.e. at the 
molecular level) crystal surface which will allow continuous growth to proceed. 
The growth will be diffusion-controlled and the face growth rates, v, will be 
linear with respect to the supersaturation, a, i.e. 

v oc a (6.46) 

For a > 5, a smooth surface is indicated and, as the high energy barrier 
discourages surface nucleation at low supersaturation, growth generally pro- 
ceeds by the screw dislocation (BCF) mechanism (equation 6.10) in which case 
the face growth rate, v, is given by 

v oc a 2 tanhCBVcr) (6.47) 

which, at low supersaturation, reduces to 

v ex a 2 (6.48) 

and at high supersaturation, to equation 6.46. 

For a values between about 2 and 5, the most probable mode of growth is the 
generation and spreading of surface nuclei, i.e. by the B + S model (section 
6.15), when equation 6.40 applies. 

For practical correlations of experimental data, however, the simple power 

v oc a r (6.49) 

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234 Crystallization 

is commonly used. This represents the two limiting cases of the BCF equation 
(r = 1 and 2, respectively) and is also a good approximation for the intermedi- 
ate regime 1 < r < 2. It also makes a satisfactory approximation for the B + S 
model over a limited range of supersaturation (Garside, 1985). 

6.1.8 Crystallization from melts 

The rate of crystallization from a melt depends on the rate of heat transfer from 
the crystal face to the bulk of the liquid. As the process is generally accom- 
panied by the liberation of heat of crystallization, the surface of the crystal will 
have a slightly higher temperature than the supercooled melt. These conditions 
are shown in Figure 6.14 where the melting point of the substance is denoted by 
T* and the temperature of the bulk of the supercooled melt by T. The overall 
degree of supercooling, therefore, is T * — T. The temperature at the surface of 
the crystal, the solid-liquid interface, is denoted by T\, so the driving force for 
heat transfer across the 'stagnant' or 'effective' film of liquid close to the crystal 
face is T\ — T. The rate of heat transfer, dq/dt, can be expressed in the form of 
the equation 

^L=hA(T 1 -T) (6.50) 

where A is the area of the growing solid surface and h is a film coefficient of 
heat transfer defined by 

h = j, (6.51) 

where k is the thermal conductivity and S' is the effective film thickness for heat 
transfer. There is a distinct similarity between the form of equation 6.50 for 
heat transfer and equation 6.16 for mass transfer by diffusion. Agitation of the 
system will reduce the effective film thickness, increase the film coefficient of 
heat transfer and tend to increase the interfacial temperature, T\, to a value near 
to that of the melting point, T*. 

The rate of crystallization of a supercooled melt achieves a maximum value 
at a lower degree of supercooling, i.e. at a temperature higher than that 

Adsorption layer 

Stagnant Bulk of melt 
^ film I 

Crystal: solution interface 

Figure 6.14. Temperature gradients near the face of a crystal growing in a melt 

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Crystal growth 235 

required for maximum nucleation. The nucleation and crystallization rate 
curves are dissimilar: the former has a relatively sharp peak (Figure 5.2), the 
latter usually a rather flat one. Tamman (1925) suggested that the maximum 
rate of crystallization would occur at a melt temperature, T, given by 

T=T «_ /A^crysA (fi ^ 

where A// cryst is the heat of crystallization and c m is the mean specific heat 
capacity of the melt. 

The crystal growth rate (e.g. mass per unit time) may be expressed as a 
function of the overall temperature driving force (cf. equation 6.18), by 

dtri i 

— =K' G A(T*-TY (6.53) 

where A is the crystal surface area, Kq is an overall mass transfer coefficient for 
growth and exponent g' generally has a value in the range 1 .5 to 2.5. The reverse 
process of melting, like that of dissolution, is often assumed to be first-order 
with respect to the temperature driving force, but this is not always the case 
(Palermo, 1967; Strickland-Constable, 1968; Kirwan and Pigford, 1969), i.e. 

--t-=K m A(T-T*) x (6.54) 


where x > \,Kyi is an overall mass transfer coefficient for melting. 
Melting is a simultaneous heat and mass transfer process, i.e. 

^ = UmAAT = - ^ AH f (6.55) 

d? d? 


dm UmAAT 

~^ = ^hT (6 - 56) 

where AT = T — T*, q is a heat quantity, AHf is the enthalpy of fusion and 
C/m is an overall heat transfer coefficient for melting. 

The surface area of the melting solid (A = (3L 2 ) is related to the mass 
(m = apL^) by 

/ \2/3 

A = /3[ — ) (6.57) 


where L is a linear dimension, p = density, and a and (3 = volume and surface 
shape factors, respectively (see section 2.14.3). Hence, equation 6.56 becomes 

and, assuming the C/m, AT, a and (3 remain constant, 

m~ 2l3 dm = ^75 / d? 

(ap) 2l3 AH{ 

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236 Crystallization 


A(m 1/3 ) = - 7 * (6.59) 

where 7 = /3(/ M AT'/3A//f(ap) 2/3 , or in terms of the change in particle size, AL, 

AL = -it (6.60) 

where 7' = f3U M AT j3AH ? ap. 

For an account of the basic theories of melting and crystal growth from 
the melt, reference should be made to the monographs by Brice (1965) and 
Ubbelohde (1965). Good accounts of dendritic growth from the melt are given 
by Gill (1989) and Ananth and Gill (1991). 

6.2 Growth rate measurements 

Many different experimental techniques have been employed to facilitate crys- 
tal growth rate measurements. The single crystal growth techniques, which can 
focus on individual face growth rates, are predominantly used for fundamental 
studies relating to growth mechanisms. Measurements made on populations of 
crystals are useful for determining overall mass transfer rates under controlled 
conditions and for observing size-dependent growth or growth rate dispersion. 
Additionally, the population methods can provide useful information for 
crystallizer design (Chapter 9). 

6.2.1 Crystal growth rate expressions 

There is no simple or generally accepted method of expressing the rate of 
growth of a crystal, since it has a complex dependence on temperature, super- 
saturation, size, habit, system turbulence, and so on. However, for carefully 
defined conditions crystal growth rates may be expressed as a mass deposition 
rate Rq (kgm~ 2 s~'), a mean linear velocity v(ms~') or an overall linear 
growth rate G (ms _1 ). The relationships between these quantities are 

. „ 1 dm 3a 
R G = K G A<* = - — = -■ Pc G 

3a dL 6a dr 6a 

= T Pc d7 = y Pc d7=y^ ( } 

where L is some characteristic size of the crystal, e.g. the equivalent sieve 
aperture size, r is the radius corresponding to the equivalent sphere, and p c is 
the crystal density. The volume and surface shape factors, a and (3, respectively, 
are defined (see section 2.14.3) by m = ap c L 3 (i.e. dm = 3ap c L 2 dL) and 
A = (5L 2 , where m and A are the particle mass and area. For spheres and cubes 
6a/ P = 1. For octahedra 6aj(3 = 0.816. 

The utility of the overall linear growth rate, G, in the design of crystallizers is 
demonstrated in section 8.3.2. Some typical values of the mean linear growth 
velocity v (= jG) are given in Table 6.1. 

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Crystal growth 237 

Table 6.1. Some mean overall crystal growth rates expressed as a linear velocity 

Crystallizing substance 


v (ms ) 

(NH 4 ) 2 S0 4 • A1 2 (S0 4 ) 3 • 24H 2 


(NH 4 ) 2 S0 4 

NH 4 H 2 P0 4 

MgS0 4 • 7H 2 

NiS0 4 • (NH 4 ) 2 S0 4 • 6H 2 

K 2 S0 4 • A1 2 (S0 4 ) 3 • 24H 2 



K 2 S0 4 

KH 2 P0 4 


Na 2 S 2 3 • 5H 2 

Citric acid monohydrate 



1.1 x 10~ 8 



1.3 x 10~ 8 



1.0 x 10- 7 



1.2 x 10~ 7 



8.5 x 10- 7 



2.5 x 10- 7 



4.0 x 10~ 7 



3.0 x 10~ 8 



6.5 x 10~ 8 



3.0 x 10~ 8 



1.1 x 10- 7 



7.0 x 10~ 8 



4.5 x 10~ 8 



8.0 x 10~ 8 



1.5 x 10- 7 



5.2 x 10- 9 



2.6 x 10~ 8 



4.0 x 10~ 8 



1.4 x 10~ 8 



2.8 x 10~ 8 



1.4 x 10- 7 



5.6 x 10~ 8 



2.0 x 10~ 7 



6.0 x 10~ 7 



4.5 x 10~ 8 



1.5 x 10~ 7 



2.8 x 10- 8 



1.4 x 10~ 7 



4.2 x 10~ 8 



7.0 x 10- 8 



3.2 x 10- 7 



3.0 x 10- 8 



2.9 x 10~ 7 



5.0 x 10- 8 



4.8 x 10- 7 



2.5 x 10- 8 



6.5 x 10- 8 



9.0 x 10~ 8 



1.5 x 10- 7 



1.1 x 10- 7 



5.0 x 10- 7 



3.0 x 10- 8 



1.0 x 10- 8 



4.0 x 10- 8 

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Table 6.1. (Continued) 

Crystallizing substance 


v (m s ) 



1.1 x lfr 8 


2.1 x 1(T 8 


9.5 x lfr 8 


1.5 x 1(T 7 

1 The supersaturation is expressed by S = c/c* with c and c* as kg of crystallizing substance per kg 
of free water. The significance of the mean linear growth velocity, v (= \G), is explained by 
equation 6.61 and the values recorded here refer to crystals in the approximate size range 0.5-1 mm 
growing in the presence of other crystals. An asterisk (*) denotes that the growth rate is probably 
size dependent. 

6.2.2 Face growth rates 

The different faces of a crystal grow at different rates under identical environ- 
mental conditions, as first demonstrated by Bentivoglio (1927). In general, the 
high index faces grow faster than the low. A fundamental assessment of the 
growth kinetics, therefore, must involve a study of the individual face growth 

An apparatus that permits precise measurement of crystal growth rates is 
shown in Figure 6.15 (Mullin and Amatavivadhana, 1967; Mullin and Garside, 
1967). Briefly, the technique is as follows. A small crystal (2-5 mm) is mounted 
on a 1 mm tungsten wire in a chosen orientation. Solution of known temper- 
ature (±0.05 °C), supersaturation and velocity is pumped through the cell, and 
the rate of advance of the chosen crystal face is observed through a travelling 
microscope. Several glass cells have been used ranging in internal diameter 
from 10 to 30 mm, permitting a wide range of solution velocities to be used. 







Figure 6.15. Single-crystal growth cell: (a) complete circuit, (b) the cell. A, solution 
reservoir; B, thermostat bath; C, thermometer; D, flow meter; E, cell; F, pump. (After 
Mullin and Amatavivadhana, 1967; Mullin and Garside, 1967) 

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Crystal growth 


20xl0" 8 





40xi0" 5 8 0xi0" 3 I2.0xl0~ 3 

Ac, concentration difference (kg of hydrate/ 
kg of solution) 

Figure 6.16. Face growth rates of single crystals of potash alum at 32°C Solution 
velocities: • = 0.127, O = 0.120, ▲ = 0.064, A = 0.022, ■ = 0.006m s" 1 . (After Mullin 
and Garside, 1967) 





The results in Figure 6.16 show the effects of both solution supersaturation 
and velocity on the linear growth rates of the (111) faces of potash alum crystals 
at 32 °C. This hydrated salt [K 2 S0 4 • A1 2 (S0 4 ) 3 • 24H 2 0] grows as almost per- 
fect octahedra, i.e. eight (111) faces. 

Three interesting points may be noted. First, the growth rate is not first-order 
with respect to the supersaturation (concentration driving force, Ac). If the 
data are plotted on logarithmic co-ordinates (not given here) straight line 
correlations are obtained giving 




where g varies from about 1.4 to 1.6. For v expressed in ms~' and Ac in kg of 
hydrate per kg of solution, K varies from about 3 x 10~ 5 to 2 x 10~ 4 as the 
solution velocity increases from 6 to 22 cm s~ ' . Second, the solution velocity has 
a significant effect on the growth rate. Third, significant crystal growth does 
not appear to commence until a certain level of supersaturation is exceeded. 

The effect of solution velocity can be seen more clearly in Figure 6.17. The 
points on this graph have been taken from the smoothed curves in Figure 6.16. 
For a given supersaturation the growth rate increases with solution velocity, the 
effect being more pronounced at the higher values of Ac. 

If the solution velocity is sufficiently high, the overall growth rate should be 
determined by the rate of integration of the solute molecules into the crystal 
lattice. If the crystal is grown in a stagnant solution (u = 0), then the rate of the 
diffusion step will be at a minimum. The growth curves in Figure 6.17 have 
therefore been extrapolated to u = and oo to obtain an estimate of the growth 
rates when the rate-controlling process is one of natural convection (u = 0) and 

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240 Crystallization 
2C1IO" 8 





_5 4x| 

0.04 0.08 0.12 0.16 0.20 0.24 oo 

u, solution velocity (m/s) 

Figure 6.17. Effect of solution velocity on the (111) face growth rate of potash alum 
crystals at 32 °C. (After Mullin and Garside, 1967) 

20*I0- 8 

40xiC~ 3 80»I0" 3 I2.0xl0" 3 

Ac, concentrotion difference ( kg of hydrote / 
Kg of solution) 

Figure 6.18. Extrapolated growth rates of potash alum crystals at limiting velocities 
(O = u — > oo, • = u — > 0) (After Mullin and Garside, 1967) 

surface reaction (u —> oo). It is, of course, unlikely that the growth curves would 
change in a smooth continuous manner when the rate-controlling mechanism 
changes from surface reaction control to natural convective diffusion control, and 
it is by no means certain that these curves can be extrapolated, with any precision, 
to the point where the growth rate becomes constant. However, the derived 
curves in Figure 6.18 give an indication of the possible limits of the growth curves. 

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Crystal growth 241 

Rates of mass transfer by natural convection are usually correlated by a semi- 
theoretical equation of the form 

Sh = 2 + a(Gr ■ Scf 25 (6.63) 

where the Sherwood number Sh = kL/D. Schmidt number Sc = r)/p s D and 
Grashof number Gr = L^p s Ap s g/ri 2 , p s = solution density, Ap s = difference 
between solution density at the interface and in the bulk solution, r\ = viscosity 
(kgm^'s -1 ),/. = crystal size (m), D = diffusivity (m 2 s _1 ) and k = a mass 
transfer coefficient (ms _1 ). The mean value of the constant a based on the 
results of a number of workers is 0.56. Growth rates, calculated from equation 
6.63 lie very close to the experimental points and this tends to confirm that the 
growth process in stagnant solution is controlled by natural convection. It is of 
interest to note in this connection that the Grashof number contains a term 
A^, which is directly proportional to the concentration difference, Ac, so the 
mass transfer rate under conditions of natural convection depends on Ac 125 . 
When forced convection is the rate-controlling process, the mass transfer rate is 
directly proportional to Ac. 

Diffusional mass transfer rates under conditions of forced convection may be 
correlated by an equation of the form 

Sh = 2 + cj>Re a v Sc b (6.64) 

where Re p is the particle Reynolds number {p^uLjrj). Equation 6.64 is fre- 
quently referred to as the Frossling equation. However, for reasonably high 
values of Sh (say > 100) it is common practice to ignore the constant 2 (the 
limiting value of Sh as Re p — > 0, i.e. mass transfer in the absence of natural 
convection) and use the simpler expression 

Sh = (j)Re a p Sc h (6.65) 

Dissolution rate data, for example, are conveniently expressed in this way. The 
mass transfer coefficient, in the Sherwood number, depends on the solution 
velocity, u, raised to the power a. It is possible, therefore, that the effect of 
solution velocity on crystal growth may also be represented by an equation of 
this type in the region where diffusion influences the growth rate. 

The effect of the two variables, Ac and u, on crystal face growth rates may 
thus be represented by 

v ha = Cu a A<? (6.66) 

where c is a constant, and a and g are both functions of the solution velocity. 
For the growth of potash alum at 32 °C, as u — > oo, u — > and g — > 1 .62, while 
as u — > 0,g — > 1.25. 

It is of interest at this point to refer back to the consequences of the BCF 
growth theory (equation 6.10). At low supersaturation, S, the growth rate is 
expected to be proportional to (S — 1) , but at high supersaturation the rate 
tends to become a linear function of S — 1. For growth from solution Chernov 
(1961) showed that for the range 1.01 < S < 1.2 (which corresponds to 

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242 Crystallization 

0.0015 < Ac < 0.03 kg/kg in the present case of potash alum) the growth rate 
can be represented by 

v m = K(S - If (6.67) 

where values of K and g are determined by the parameters in the theoretical 
relationship. Now S = c/c*, i.e. S — 1 = Ac/c*, so equations 6.62 and 6.67 are 
of the same form and the value of the exponent g, measured experimentally, is 
within the range predicted theoretically. 

The velocity of the solution past the crystal face is thus capable of influen- 
cing the growth rate, and this velocity effect manifests itself as a crystal size 
effect when freely suspended crystals are grown in a crystallizer. The reason, of 
course, is that large crystals have higher settling velocities than small crystals, 
i.e. higher relative solid-liquid velocities are needed to keep the larger crystals 
suspended. This important effect, which has not often been appreciated in the 
past, can rapidly be detected and quantified in the growth cell described 

Salts that have been established as having solution velocity dependent 
growth rates include ammonium and potassium alums, nickel ammonium 
sulphate, sodium thiosulphate and potassium sulphate. Ammonium sulphate, 
ammonium and potassium dihydrogen phosphates, for example, do not. 

6.2.3 Layer growth rates 

The movement of growth layers on the face of a crystal growing in solution can 
often be detected and measured by observing the particular face microscopic- 
ally, using reflected light. An apparatus that permits this to be done is shown in 
Figure 6.19. Small crystals are nucleated and grown on the lower non-reflecting 
surface of the observation cell. 

The arrangement consists of a central portion, 20 mm in diameter and 4 mm 
deep, in which the crystals are growing, enclosed in a water jacket which 
controls the cell temperature to within ±0.05 °C. Solution is circulated through 
the cell under controlled conditions of temperature, supersaturation, flow rate 
and purity. The solution velocity across the central portion of the cell may be 
varied between about 1 and 20mms~ 1 . The crystals growing in the cell are 
illuminated with a highly collimated, intense light beam from a 24-V, 150-W 
tungsten-halogen lamp. Angular adjustment of the cell in the horizontal and 
vertical planes allows light reflected from the crystal surface to be diverted into 
the microscope objective. The growth layer velocities are measured with the aid 
of a micrometer eyepiece. 

Extensive use of this type of cell for the measurement of layer velocities on 
crystal faces has been reported by Davey and Mullin (1974). The moving layer 
fronts observed by this technique are not elementary (monomolecular) steps 
but macrosteps, often several hundred or thousand molecules in height, which 
build-up from the bunching of smaller layers with velocities a hundred times 
faster than the macrosteps (Phillips and Mullin, 1976). These fast moving layers 
are generally difficult to monitor, but velocity measurements of near-elementary 

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Crystal growth 243 

Glass window 
Rubber flange \ Aluminium O-rings 

Water in 

v t ' - ' ' ■ '- ' . ' - ' ' " '1 l ' ~J f f 

Solution in 

Figure 6.19. A cell for making layer growth observations hy reflection microscopy. (After 
Davey and Mullin, 1974) 

layers have recently been made using atomic force microscopy, a powerful tool 
that promises to cast new light on the fundamental mechanisms of crystal 
growth (Land et al, 1999). 

6.2.4 Overall growth rates 

It is often much more convenient, and more useful for crystallizer design 
purposes, to measure crystal growth rates in terms of mass deposited per unit 
time per unit area of crystal surface rather than as individual face growth rates. 
This may be done in agitated vessels or fluidized beds, e.g. by measuring the 
mass deposition on a known mass of sized seed crystals under carefully con- 
trolled conditions. 

The overall linear growth rate, G, (ms _1 ) may then be evaluated from 








where M\ and Aff are the initial and final crystal masses (kg), respectively. N is 
the number of individual crystals, a is their volume shape factor, p is their 
density (kgm~ 3 ) and t is time (s). 

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244 Crystallization 

Alternatively, expressing the overall mass deposition rate Rq (kgm~ 2 s _1 ) as 
in equations 6.14 and 6.61 the overall linear growth rate G may be expressed as 

G = —K G Ac s (6.69) 


where (3 is the crystal surface shape factor and Ac is the mean supersaturation 
over the run (kg solute/kg solvent). The value of exponent g is given by the slope 
of the linear plot of log G versus log Ac, and the overall mass transfer coeffi- 
cient Kq (kgm~ 2 s _1 ) can then be evaluated. 

Experimental precautions 

A number of precautions need to be taken when attempting to measure reliable 
crystal growth rates by gravimetric measurement and subsequent calculation 
(Phillips, 1974). The seed crystals should be carefully selected, both for size and 
surface quality. The crystal surface area is often required and this is most 
commonly calculated on the assumption that the crystals have a definite 
geometrical form and plane faces. The seeds ought to have faces that, macro- 
scopically at least, are smooth. The volume and surface shape factors of the 
seeds and the grown crystals should be determined (section 2.14.3) so that any 
changes may be taken into account in subsequent calculations, e.g. when using 
equations 6.68 or 6.69. The volume shape factor of potassium sulphate crystals, 
for example, changes from about 1 to 0.6 as the crystals grow from about 
300 um to 2mm (Mullin and Gaska, 1969; Garside, Mullin and Das, 1973). 

At the end of a growth run, the crystals must be cleanly and qualitatively 
separated from the mother liquor so that their final dry mass can be measured. 
Filtration is commonly followed by washing to recover residual mother liquor. 
These operations should be carried out rapidly to minimize any chance of the 
crystals undergoing change. Ideally the wash liquid should be completely mis- 
cible with the mother liquor, and the crystals should be practically insoluble in 
the wash liquid. Further, to assist rapid drying, the wash liquid should be 
reasonably volatile. Methanol, ethanol and acetone, for example, are often 
chosen for inorganic salts that have crystallized from aqueous solution. Filtra- 
tion should remove a very high proportion of the mother liquor so that the 
chance of salting-out and consequent surface contamination is minimized. 
Further comments on the problems associated with crystal washing under 
industrial conditions are made in section 9.7.1. 

Fluidized beds 

A laboratory-scale fluidized bed crystallizer capable of yielding useful growth 
rate information is shown in Figure 6.20. It is constructed mainly of glass (total 
capacity 10-13 L) with growth zones 5-8 cm diam. and 75 cm long. A combina- 
tion of heating tapes and water cooler enables the temperature of the solution 
to be maintained to ±0.03 °C. Solution concentration can be measured at 
intervals or continuously. A typical run would consist of adding about 

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Crystal growth 245 

ac Ch #> ar ~ 

Figure 6.20. A laboratory-scale jluidized bed crystallizer: A, growth zone; B, outlet cock; 
C, resistance thermometer; E, F, orifice plates; H, heating tapes; J, thermometer; K, water 
cooler. (After Mullin and Garside, 1967) 

5 g ± 1 mg of carefully sized seed crystals and controlling the solution velocity 
so that the crystals are uniformly suspended in the growth zone until their mass 
has increased to, say, 15g. This mass increase would allow 600 um crystals of 
potassium sulphate, for example, to grow to about 800 urn. The duration of 
a run varies from about | to 3 h, depending on the working level of super- 
saturation. At the end of a run the crystals are removed, dried, weighed and 
sieved. Some typical results are shown in Figure 6.21 for potash alum crystals 
grown at 32 °C. These results may be compared with those shown in Figure 
6.16. Here, again, the effect of supersaturation can be seen and so can the effect 
of crystal size. As explained above, solution velocity dependent growth shows 
up as crystal size dependent growth when freely suspended crystals are grown in 
a crystallizer. In this case large crystals grow faster than small. 
For potash alum it has been shown that 

R G = K G Ac l 


For Rq expressed as kgm~ 2 s~' and Ac as kg hydrate/kg solution Kq varies 
from 0.115 to 0.218 and g from 1.54 to 1.6 for crystals ranging from 0.5 to 
1.5 mm. Or, since it has already been shown that v^u = Cu a Ac g for single 
crystals (equation 6.66), then 


C'L m Ac g 


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4 x 1(T 3 8 x 1CT 3 12 x 10 _a 16 x 10 -3 20 x 1CT 3 

Ac, concentration _. , 

kg of solution) 

difference (kg of hydrate/ 

Figure 6.21. Overall growth rates of potash alum crystals at 32 °C (mean crystal sizes: 
A = 1.96, ■= 1.4, = 0.99, A=0.75, v = 0.55mm. (After Mullin and Garside, 1967) 

which is similar to the equation used by Bransom (1960) as the starting point for a 
theoretical analysis of crystal size distribution. In the above case of potash alum 


16Z 063 A 


Overall growth rates for potash alum measured in the fluidized bed crystal- 
lizer coincide very well with those predicted from face growth rates measured in 
the single crystal cell (Figure 6.22). The alums grow as almost perfect octa- 
hedra, i.e. eight (1 1 1) faces, so it is a simple matter, using the crystal density, p c , 
to convert linear face velocities to overall mass deposition rates (Rq = Pc^ni))- 

Agitated vessels 

It is possible to determine overall crystal growth rates by adding a known mass 
of sized seeds to a supersaturated solution in an agitated vessel, following 
a similar procedure to that outlined above for the fluidized bed method. To 
correlate the data, however, it is necessary to estimate the particle-fluid slip 
velocity as a function of impeller speed in the agitated vessel using relationships 
of the type described in section 9.4.1. 

An example of the comparison of growth rate data obtained in both fluidized 
bed and agitated vessel crystallizers, using ammonium alum, has been reported 
by Nienow, Bujac and Mullin (1972). 

Measurement from desupersaturation rates 

A rapid method for overall crystal growth rate estimation may be made by 
suspending a batch of seed crystals in a supersaturated solution kept at con- 
stant temperature, and following the decay of supersaturation over a period 
of time. A mass of seed crystals of known size and surface area is added to 
the solution in a closed system, e.g. in a fluidized bed or an agitated vessel. 

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Crystal growth 


20«lO" 8 

E 16 x 10" 




Z 12 « 10 



9 8xiO 



c -a 

A C =0.009 

/ _, 



A c =0.006 
A <r_=0003 

04 08 12 0.16 0.20 24 
Relative velocity between crystal and solution, 

Figure 6.22. Comparison between face {smooth curve) and overall (points) growth rates of 
potash alum crystals at 32°C. (After Mullin and Garside, 1967) 

The initial supersaturation is recorded and the desupersaturation decay 
is monitored by continuous or frequent intermittent solution analysis, e.g. by 
measuring some relevant physical property such as density, refractive index, 
conductivity, etc. The same procedure may be used, with appropriate nomencla- 
ture changes, to determine overall dissolution rates by measuring the increase in 
solution concentration. 

Assuming that negligible nucleation occurs after the seeds are added, the 
change in solution concentration dc at any instant is proportional to the mass 
deposition dm on existing crystals, i.e., 




Wis the mass of 'free water' present. If the crystallizing substance is hydrate, the 
solution concentration c should be expressed as kg of hydrate per kg of free water. 
The overall crystal growth rate, Rq (the mass rate of deposition, dm/dt, per 
unit crystal surface area, A, see equation 6.61) may thus be expressed as 

R ° = A 


W dc 


Values of dc/dt may be obtained from the measured desupersaturation curve. 
W is a constant for a given run and the surface area, A, of the added seeds can 
be estimated from their total mass, M, and characteristic size, L: 

(3Mjap c L 


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248 Crystallization 

where a and j3 are the volume and surface shape factors, respectively, and p c 
is the crystal density. It is advisable to use seed crystals as near uniform as 
possible, in both size and shape, to minimize errors. 

If the surface area change cannot be neglected, a mean value A may be used 
in equation 6.75, based on the initial and final areas, and calculated from (Ang 
and Mullin, 1979) 

A = l - f A (—) &L (6.76) 

L t - L Q J, \L 


MA 113 (MA 2 ' 3 

M ) ' \M J 


where Ma and M t are the initial and final crystal masses. 

Desupersaturation methods for crystal growth rate measurements have been 
reported for ammonium alum (Bujac and Mullin, 1969), potassium sulphate 
(Jones and Mullin, 1973a), nickel ammonium sulphate (Ang and Mullin, 1979), 
potassium chloride (Nyvlt, 1989) and succinic acid (Qui and Rasmuson, 1990). 

A different approach was adopted by Garside, Gibilaro and Tavare (1982) 
who suggested that crystal growth rates could be evaluated from a knowledge 
of the first two zero-time derivatives of a desupersaturation curve which had 
been approximated by an «th order polynomial. The analytical procedures 
adopted are fully described in the above paper, together with an example of 
the application of the approach to the growth of potassium sulphate crystals in 
a fluidized bed crystallizer. 

Measurement on a rotating disc 

The rotating disc method may be used to study the separate roles of diffusion 
and integration in crystal growth since it enables the mass transfer (diffusion) 
step to be isolated. A uniform hydrodynamic boundary layer of thickness 

<5 h = 2.8(^/w) 1/2 (6.78) 

is produced over the smooth surface of a small disc rotated in a horizontal plane 
about its axis; v is the kinematic viscosity of the liquid and to is the angular 
velocity of the disc. For example, a disc rotating at N = 200 rev/ min (ui = 
2ttN ~ 21 radians/s), and takings = 10 _6 m 2 s _1 , gives the value of 8^ ~ 600 um. 
The mass transfer (diffusion) boundary layer thickness <5 m would only be a small 
fraction of this (see section 6.3.2). 

The disc, impregnated with the crystalline material, is rotated in a relatively 
large volume of solution so that the solution concentration remains virtually 
unchanged during a run. 

The technique may be used to study both growth and dissolution using 
solutions of the appropriate solute concentrations. It has been used to measure 
individual face growth rates by mounting a well-formed crystal in the disc with 
one face only exposed, but it is more commonly employed for measuring 
overall growth or dissolution rates of a multicrystalline compact compressed 

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Crystal growth 249 

into a recess in the disc surface, thus exposing a random orientation of faces to 
the solution. 

Flow over the disc must be laminar, so the disc Reynolds number 
(Re = Ljr 2 jv) should be kept between 10 3 and 10 5 . For example, for a 50-mm 
diameter disc, these limits correspond to rotational speeds of around 1 5 and 
1500rev/min, respectively. Turbulent flow starts at a Reynolds number of 
about 2 x 10 5 and below about 10 2 natural convection can interfere with the 
mass transfer process. The disc is weighed before and after a run during which 
a loss, or gain, of around 0.2 g in mass has occurred, depending on whether 
growth or dissolution is being studied. It is essential, of course, to standardize 
the disc-drying procedure in such studies. 

Descriptive accounts of the construction and use of rotating disc units have 
been given by Bourne et al. (1976), Karel and Nyvlt (1989) and Garside, 
Mersmann and Nyvlt (1990). 

6.2.5 Growth and nucleation rates 

The processes of growth and nucleation interact in a crystallizer, and both 
contribute to the crystal size distribution (CSD) of the product (see section 9.1). 
Kinetic data needed for crystallizer design purposes (effective growth and 
nucleation rates) can be conveniently measured on the laboratory scale in 
a mixed-suspension, mixed-product removal (MSMPR) crystallizer operated 
continuously in the steady state (Figure 9.3). The assumptions made are that no 
crystals are present in the feed stream, that all crystals are of the same shape, 
that crystals do not break down by attrition, and that crystal growth rate is 
independent of crystal size. 

The relationship between crystal size, L, and population density, n (number 
of crystals per unit size per unit volume of the system), derived directly from the 
population balance (Randolph and Larson, 1988) (section 9.1.1) is 

n = H exp(-L/Gr) (6.79) 

where «o is the population density of nuclei (zero-sized crystals) and r is the 
residence time. Equation 6.79 describes the crystal size distribution for steady- 
state operation. Rates of nucleation B and growth G(= dLjAt) are convention- 
ally written in terms of supersaturation as 

B=k { Ae h (6.80) 


G = k 2 Ac g (6.81) 

These empirical expressions can be combined to give 

B = k^G 1 (6.82) 


i = b/g (6.83) 

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in which b and g are the kinetic orders of nucleation and growth, respectively, 
and i is the relative kinetic order. The relationship between nucleation and 
growth may be expressed as 

B = n G 



«o = k/[G 



Experimental measurement of crystal size distribution (recorded on a num- 
ber basis) in a steady-state MSMPR crystallizer can thus be used to quantify 
nucleation and growth rates. A plot of log n vs. L should give a straight line of 
slope -{Gt)~ with an intercept at L = equal to «o (equation 6.79 and Figure 
6.23a); if the residence time r is known, the crystal growth rate G can be 
calculated. Similarly, a plot of log«o vs. logG should give a straight line of 
slope i — 1 (equation 6.85 and Figure 6.23b); if the order g of the growth process 
is known, the order of nucleation b can be calculated from equation 6.83. 

A typical laboratory MSMPR crystallizer suitable for measuring kinetic 
data is shown in Figure 6.24. Such a unit would typically be operated for 
around ten residence times to achieve the steady-state conditions necessary 
before taking a sample of the magma to assess the crystal size distribution. 
The solenoid-operated discharge mechanism is based on the one described by 
Zacek et al. (1982). Normally only one feed system would be required, e.g. for 
cooling crystallization, but two independent feed systems as illustrated, would 
be necessary for reaction crystallization or precipitation studies. With suitable 
modification to the crystallization vessel, the unit can be adapted for reduced- 
pressure evaporation. 

MSMPR units with crystallizer working volumes as small as 250 mL have 
been operated successfully, but if the kinetic data are to be used for industrial 
design purposes, the working volume should not be less than about 4L, and 
sizes up to 20 L have been recommended (Garside, Mersmann and Nyvlt, 

Crystal size, L 

Growth rate G ( log scale) 

Figure 6.23. Population plots characterizing (a) the crystal size distribution and (b) the 
nucleation and growth kinetics for a continuous MSMPR crystallizer 

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Crystal growth 


Figure 6.24. A laboratory-scale MSMPR crystallizer: A, thermostatted feedstock tank; 
B, constant-head tank; C, MSMPR crystallizer; D, water inlet to jacket; E, baffle; 
F, thermometer; G, level detector; H, solenoid-operated discharge valve; I, magma outlet; 
J, control unit 

1990). The larger the working volume, the more meaningful will be the nucle- 
ation data, particularly for scale-up purposes, but the more difficult it will be to 
achieve good mixing and MSMPR conditions in the vessel. Further, because of 
the large quantities of feedstock solution to be handled, more expensive ancil- 
lary equipment will be required. Conversely, although it is much easier to 
achieve MSMPR operation in small volume units and to operate with much 
simpler equipment, the consequent low feedstock solution flowrates in narrow 
supply lines can cause severe problems arising from crystallization blockage. 
A detailed example of the evaluation of kinetic information from MSMPR data 
is given in section 9.2. 

Mersmann and Kind (1988) have surveyed data reported in the literature on 
17 different inorganic substances crystallizing or precipitating from aqueous 
solution in MSMPR crystallizers. One of the interesting compilations is shown 
in Figure 6.25 where some orders of magnitude of potential growth and 
nucleation rate are indicated. Below a relative supersaturation, a{= Ac/c*), 
of about 1, the processes could be described as crystallization (by cooling, 
evaporation, salting out, etc.) coupled with secondary nucleation. For a > 1 
the processes are more appropriately described as precipitation coupled with 
primary nucleation. 

6.2.6 Effect of temperature 

The relationship between a reaction rate constant, k, and the absolute temper- 
ature, T, is given by the Arrhenius equation 




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IO 7 

IO 8 


I l0 " 


Nucleation rate, B, m -3 s _l 

IO 9 IO 10 10" IO 12 

-1 1 1 





(primary nucleation) 

(secondary nucleation 




I0 U 

Relative supersaturation, a 

Figure 6.25. Growth and nucleation rates in an MSMPR crystallizer: A, KC1; B, NaCl; 
C, (NH 2 ) 2 CS; D, (NH 4 ) 2 S0 4 ; E, KN0 3 ; F, Na 2 S0 4 ; G, K 2 S0 4 ; H, NH 4 A1(S0 4 ) 2 ; 
/, K 2 Cr 2 O v ; /, KA1(S0 4 ) 2 ; N, CaC0 3 ; 0, Ti0 2 ; Q, BaS0 4 . (After Mersmann and Kind, 

where E is the energy of activation for the particular reaction. On integration 
equation 6.86 gives 

k = A-exp(-E/RT) 

or, taking logarithms, 


In k = In A 



Therefore, if the Arrhenius equation applies, a plot of log k against J" -1 should 
give a straight line of slope —E/R and intercept log A:. 

Alternatively, if only two measurements of the rate constant are available, k\ 
at T\ and k 2 at 7*2, the following equation may be used: 

RT,T 2 k 2 

T 2 - T x k x 


Equation 6.89 is obtained by integrating equation 6.86 between the limits T\ 
and T 2 assuming that E remains constant over this temperature range. 

The above equations may be applied to diffusion, dissolution or crystal- 
lization processes; k can be taken as the relevant rate constant. For example, 
a plot of log^o versus T~ l would give a so-called activation energy for crystal 

growth, ^cryst; \ogK D versus T gives E disi 
D = diffusivity, gives -Ediff; and log?? versus T 
a value of ,/ivisc; and so on. 

Activation energies for diffusion are usually r. 
face integration ~40— 60kJmol~ . As the rate of integration increases more 

logZ) versus T ', where 
1 , where r/ = viscosity, gives 

10— 20kJmol _1 and for sur- 

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Crystal growth 253 

rapidly with temperature than does the rate of diffusion, crystal growth rates 
tend to become diffusion controlled at high temperature and integration con- 
trolled at low temperature. For example, sucrose is reported to be diffusion 
controlled above 40 °C (Smythe, 1967) and sodium chloride above 50 °C (Rum- 
ford and Bain, 1960). Over a significant intermediate range of temperature, 
however, both processes can be influential, and accordingly Arrhenius plots of 
crystal growth data often give curves rather than straight lines, indicating that 
the apparent activation energy of the overall growth process is temperature 

6.2.7 Effect of crystal size 

It is probably true to say that all crystal growth rates are particle size depend- 
ent; it all depends on the size and size range under consideration. The effect of 
size may be quite insignificant for macrocrystals, but the situation can change 
dramatically for crystals of microscopic or sub-microscopic size. 

Size-dependent growth 

One effect of crystal size on the overall growth rates of macrocrystals has 
already been mentioned in section 6.2.4 (see Figure 6.21). Not all substances 
exhibit this type of size-growth effect, but in cases where they do, an overall 
growth rate expression of the form of equation 6.71 can be useful. Because of 
the limitations imposed by traditional experimental techniques, the crystals 
normally studied do not extend much outside the range 200 nm to 2 mm. In 
this range any effect of size would appear to be closely linked with the effect of 
solution velocity: large particles have higher terminal velocities than those of 
small particles and, in cases where diffusion plays a dominant role in the growth 
process, the larger the crystals the higher their growth rate. 

A different effect may be considered for crystals smaller than about 10 um. 
Because of their very small terminal velocity, and sizes smaller than that of 
turbulent eddies, they may be growing in a virtually stagnant medium, even in 
an apparently well-agitated system. 

Another, and often more powerful, effect of crystal size may be exhibited at 
sizes smaller than a few micrometres, and is caused by the Gibbs-Thomson 
effect (section 3.7). Crystals of near-nucleic size may grow at extremely slow 
rates because of the lower supersaturation they experience owing to their higher 
solubility. Hence the smaller the crystals, in the size range below say 1 or 2 um, 
the lower their growth rate. 

A third factor to be considered in connection with the crystal size-growth 
rate effect is the possibility of the surface integration kinetics being size depend- 
ent. The number of dislocations in a crystal increases with size due to mechan- 
ical stresses, incorporation of impurity species into the lattice, etc. In addition, 
the larger the crystals the more energetically will they collide in agitated 
suspensions and the greater is the potential for surface damage. Both of these 
effects favour faster surface integration kinetics and lead to higher growth rates 
with increasing crystal size (Garside and Davey, 1980). 

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254 Crystallization 

Some comments are made in section 9.1.1 on attempts to develop appropriate 
empirical formulae to relate crystal growth rate with crystal size, particularly 
for the assessment of MSMPR crystallizer data. 

Growth rate dispersion 

The above size-dependent effects are all concerned with the growth rate change 
of a crystal solely on account of its size, i.e. a genuine size-growth effect. In 
contrast, the behaviour now generally known as 'growth rate dispersion' refers 
to the fact that individual crystals, all initially of the same size, each apparently 
subjected to identical growth environments (temperature, supersaturation, 
hydrodynamics, etc.), can grow at different rates. White and Wright (1971) 
first identified this phenomenon in the batch crystallization of sucrose, and this 
is now a generally accepted behaviour in all crystallizers. It has also been 
demonstrated for the growth of secondary nuclei (Garside, Rusli and Larson, 
1979; Berglund, Kaufman and Larson, 1983). Reviews of the subject have been 
made by Ulrich (1989) and Tavare (1991). 

Growth rate dispersion stems mainly from different interferences with the 
surface integration kinetics on different crystals. Random surface adsorption 
or physical incorporation of impurity species, leading to the development 
of different crystallographic faces, may account for some cases, but there is 
evidence to suggest that the prime causes could be the varying degrees of lattice 
strain and deformation in individual crystals and their dislocation structure 
(Ristic, Sherwood and Shripathi, 1991; Jones et al., 2000). Lattice strain can be 
caused by mechanical stresses imparted to crystals in a crystallizer by fluid 
shear, or physical contact with other crystals, the agitator or other internal 
parts of the equipment. The less ductile the crystals the more likely they are to 
be prone to growth rate dispersion. 

6.2.8 Effect of impurities 

The presence of impurities in a system can have a profound effect on the growth 
of a crystal. Some impurities can suppress growth entirely; some may enhance 
growth, while others may exert a highly selective effect, acting only on certain 
crystallographic faces and thus modifying the crystal habit (see section 6.4). 
Some impurities can exert an influence at very low concentrations, less than 
1 part per million, whereas others need to be present in fairly large amounts 
before having any effect. The influence of impurities on nucleation has been 
discussed in section 5.4. 

Any substance other than the material being crystallized can be considered 
an 'impurity', so even the solvent from which the crystals are grown is in the 
strictest sense an impurity, and it is well known that a change of solvent 
frequently results in a change of crystal habit (see section 6.4.2). 

Impurities can influence crystal growth rates in a variety of ways. They can 
change the properties of the solution (structural or otherwise) or the equilib- 
rium saturation concentration and hence the supersaturation. They can alter 
the characteristics of the adsorption layer at the crystal-solution interface and 

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Crystal growth 






Figure 6.26. Sites for impurity adsorption on a growing crystal, based on the Kossel model: 
(a) kink; (b) step; (c) ledge (face). (After Davey and Mullin, 1974) 

influence the integration of growth units. They may be built into the crystal, 
especially if there is some degree of lattice similarity. 

Impurities are often adsorbed selectively on to different crystal faces and 
retard their growth rates. To effect retardation, however, it is not necessary for 
the impurity to achieve total face coverage. As seen in Figure 6.26, utilizing the 
Kossel model (section 6.2), three sites may be considered at which impurity 
species may become adsorbed and disrupt the flow of growth layers across the 
faces, viz. at a kink, at a step or on a ledge (face) between steps. Considering the 
theoretical implications of adsorption at each of these sites in relation to 
experimental observations, it is possible to assess which of the adsorption sites 
are important in reducing layer velocities (Davey and Mullin, 1976). Briefly, if 
kink site adsorption is possible, growth retardation may be affected at very low 
impurity levels in the solution. More impurity would be needed if step site 
adsorption is the preferred mode while much higher levels may be required if 
adsorption only occurs on a ledge or face site. 

The use of single-crystal growth-rate measurements in the quantitative pre- 
diction of crystal habit was first demonstrated by Michaels and Colville (1960) 
who grew adipic acid crystals from aqueous solution in the presence of trace 
surfactants. Sodium dodecylbenzenesulphonate (SDBS) (anionic) caused a much 
greater reduction in the growth rate of the (010) and (110) faces than of the 
(001) face, leading to the formation of prisms or needles. Trimethyl dodecyl- 
ammonium chloride (TMDAC) (cationic) had the opposite effect, favouring 
the formation of plates or flakes. 

A similar study was made by Mullin and Amatavivadhana (1967) and 
Mullin, Amatavivadhana and Chakraborty (1970) on the face growth rates of 
ammonium and potassium dihydrogenphosphates which are affected by trace 

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Figure 6.27. (a) Perfect 'capping' of an ADP crystal; (b) tapered growth caused by traces 
of Fe 3+ (9 = angle of taper). (After Mullin, Amatavivadhana and Chakraborty, 1970) 

quantities of Cr 3+ or Fe 3+ and also by solution pH changes. The most visual 
effect is a tapering of the prism faces (Figure 6.27), the angle of taper increasing 
with increase in impurity cation concentration. The overall effect, which was 
confirmed in a fluidizing bed crystallizer, is that growth at pH 4 gives thin 
needles, at pH 5 gives squat prisms and at pH 4 with 5 ppm Fe 3+ in the solution 
gives tapered needles. Possible mechanisms for the complex action of these 
trivalent cations, and the effect of pH, have been proposed by Davey and 
Mullin (1976) and Kubota et al. (1994, 1999). This topic is considered further 
in section 6.4. 

Theoretical analyses of the effects of impurities on crystal growth have been 
made by Bunn (1933), Lacmann and Stranski (1958), Chernov (1965), Davey 
(1976) and Boistelle (1982). Cabrera and Vermilyea (1958) visualized a general 
impurity effect in terms of a 'pinning' mechanism whereby the progress of 
growth layers on a crystal surface is blocked by individually adsorbed impurity 
species. They proposed that complete stoppage of growth would occur when 
the distance between the adsorbed impurities species was < 2r c , where r c repre- 
sented the radius of a critical two-dimensional nucleus (equation 6.5). For 
spacings > 2r c the elementary growth layers could squeeze through the gaps 
between the impurity species and crystal growth would continue, although at a 
lower rate than that without any impurities present. 

The blockage of active sites by impurities can be related to the impurity 
concentration in solution through the Langmuir adsorption isotherm, and 
a number of models utilizing this concept have been proposed (Davey and 
Mullin, 1974; Black et al, 1986; Klug, 1993). A recent refinement, which 
incorporates the concept of an impurity effectiveness factor (Kubota and 
Mullin, 1995), offers an opportunity to explain several hitherto anomalous 
patterns of behaviour and may be summarized as follows. 

The growth layer velocity v in the presence of an impurity relative to the 
velocity vq in pure solution may be represented by 

V/VQ = 1 



where 9 eq is the fractional surface coverage by adsorbed impurities at equilib- 
rium, and a is an impurity effectiveness factor. Thus when a = 1 and 8 eq = 1, 

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Crystal growth 


the step velocity v = 0, i.e., complete stoppage of growth at complete coverage. 

However, when a > 1 and 6> eq < 1 (incomplete coverage) 
a < 1 , v never approaches zero, even for 8 eq = 1 . 

Assuming the Langmuir adsorption isotherm to apply, 

9 eq = Kc/(1 + Kc) 

0, but when 


where K is the Langmuir constant and c is the impurity concentration. The 
relative step velocity can be expressed as 

v/v = 1 - [aKc/(l + Kc)] 


and assuming that the crystal face growth rate G is proportional to the step 

G/G = 1 - [aKc/(l + Kc)] 


Relative step velocities calculated from equation 6.92 are shown in Figure 6.28 
for several different effectiveness factors as a function of the dimensionless 
impurity concentration Kc. When a > 1, the relative velocity decreases very 
rapidly with increasing impurity concentration, reaching zero at a small value 
of Kc. For a = 1, the step velocity approaches zero asymptotically. For a < 1, 
however, the step velocity never approaches a non-zero value as the impurity 
concentration is increased. These three types of behaviour in the step velocity- 
impurity relationship can be found in many reports in the literature. For 
example, the case of a > 1 is illustrated by the effect of raffinose on the step 
velocities on the {100} faces of sucrose (Albon and Dunning, 1962). The effects 
of FeCh and AICI3 on the step velocities on the {100} faces of ammonium 
dihydrogen phosphate (Davey and Mullin, 1974) are good examples of a = 1, 



~ 0.8 












1 1 1 






l i i 


5 10 15 20 25 

Dimensionless impurity concentration Kc 


Figure 6.28. Relationship between the relative step velocity vjv and the dimensionless 
impurity concentration Kcfor different values of the impurity effectiveness factor a. (After 
Kubota and Mullin, 1995) 

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258 Crystallization 

and the effect of aliphatic carboxylic acids on the {100} face growth rates of 
KBr (Bliznakov and Nikolaeva, 1967) neatly illustrates the case of a < 1. 

The impurity effectiveness factor a is related to the critical radius of a two- 
dimensional nucleus at low supersaturation (a <C 1) Mullin and Kubota 
(1995) as 

a = ja/kTaL (6.94) 

where a is the size of a growth unit, 7 the edge free energy, a the supersaturation 
(equation 3.69), L the separation of sites available for impurity adsorption, 
T the absolute temperature and k the Boltzman constant. 

Equilibrium adsorption, however, is neither necessary for impurity action, 
nor is it the most commonly encountered condition. Impurities can still retard 
growth rate under non-equilibrium adsorption conditions so long as sufficient 
surface coverage is attained. To consider non-steady-state impurity action 6> eq 
in equation 6.90 is replaced by 9, the surface coverage at time t: 

v/v = l-a0 (6.95) 

and, assuming the Langmuir mechanism to apply, the net adsorption rate can 
be expressed as 

d0/dt = ki(l - 9)c-k 2 6 (6.96) 

where k\ and k 2 are constants and c is the impurity concentration, also assumed 
to be constant. Integrating equation 6.96 with the initial condition of 9 = at 
t = for a given impurity concentration c gives the surface coverage 9 as 
a function of time: 

0=0 e q[l-exp(-r/r)] (6.97) 

where the adsorption process time constant r = (k\ + k 2 Y . The final equilib- 
rium coverage 6> eq is given by equation 6.91 and from equations 6.95 and 6.97 

v/vq = 1 - a9 eq [l - exp(-//r)] (6.98) 

or in terms of face growth rates 

G/G = 1 - a9 eq [l - exp(-f/r)] (6.99) 

Equations 6.98 and 6.99 are valid for all values of t for weak impurities (a < 1) 
and up to a characteristic time t c , when the face growth rate G becomes zero, 
for strong impurities (a > 1) where 

f c = ln[a0 eq /(a0eq-l)]T (6.100) 

The combined influence of supersaturation and impurity concentration on 
crystal growth can be quite complex, but two basic cases may be considered 
(Kubota, Yokota and Mullin, 2000): (i) growth is only suppressed in the low 
range of supersaturation while at higher supersaturations the impurity effect 
disappears completely and (ii) growth rate suppression occurs throughout 
a very wide range of supersaturation. The first case may be explained by 

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Crystal growth 


assuming slow unsteady-state adsorption of impurity at high supersaturation. 
The second is actually a special case of the first, where the adsorption time 
constant becomes very small even at higher supersaturations. 

For the simple case in which adsorption equilibrium is established instant- 
aneously (r = 0) regardless of the supersaturation, equation 6.99 reduces to the 
equilibrium adsorption model equation 




for a8 eq <C 1 

which, using equations 6.91 and 6.94, becomes 
G/G = 1 - [(Kc/1 + Kc)(ja/kFcrL)] 



Equation 6. 102 can be modified to describe the relative growth rate as a function 
of supersaturation a at a given temperature under the influence of a given 
impurity concentration 

G/G = 1 - (a/a c y 

for u,<a«l 


where <r c , the critical supersaturation below which G = 0, is defined by 

ct c = 7aKc-/k7X(l + Kc) (6. 104) 

Any growth model can be used for Go in equations 6.102 and 6.103, but if 
a linear model (Go = kacr) is assumed, equation 6.103 becomes 

koicr — <t c ) for <7 C < a <C 1 


Equation 6.105 is represented by the dotted line in Figure 6.29a showing that 
for instantaneous adsorption (r = 0) growth rate suppression occurs over a 
wide range of supersaturation. For the case of very slow adsorption (r = oo), 
no impurity effect would be expected, i.e., growth in the presence of impurity 
would be the same as if no impurity were present, i.e. G = Gq. This is repres- 
ented by the continuous line in Figure 6.29a. 

In most cases the impurity adsorption rate decreases as the supersaturation is 
increased. The time constant r increases from zero at some critical supersatura- 
tion (To, below which adsorption occurs instantaneously. The time-averaged 
growth rate would change gradually from G for r = (instantaneous adsorp- 
tion) to Go (the growth rate in pure solution) for r = oo (very slow adsorption) 


pure or impure (T^w) 



impure (T*0) *r 


pure \<ff 

^\s / .--'X 


jS Jy' impure (T=0) 

/ S 

CT C Of) 






pure \ 



' _.. 

impure (t=0) 


OS o c 

Figure 6.29. Face growth rate G as a function of supersaturation a: (a) for instantaneous 
(t = 0) and very slow (r = oo) adsorption, (h) and (c)for a continuous increase of V from 
to oo, (b) for a > a c and (c) for a < a c . (After Kuhota, Yokota and Mullin, 2000) 

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260 Crystallization 

in the manner shown schematically in Figures 6.29b and 6.29c as thick solid 
lines for the cases of a c < co and a c > <jq. 

Another interesting behaviour, often exhibited in the presence of impurities, 
is growth rate hysteresis where a crystal growing in a high supersaturation 
solution can continue to grow, at appropriate reduced rates, down to a low 
supersaturation, if the supersaturation is lowered continuously from the higher 
level. Yet the reverse does not occur, i.e., a crystal which has ceased to grow at 
a low supersaturation is unable to grow even when the supersaturation is 
continuously raised to a very much higher level. The hysteresis effect is an 
indication of unsteady-state growth behaviour and can be explained by assum- 
ing a slow impurity adsorption at higher supersaturations as discussed above. If 
the supersaturation is lowered from a high value the crystal can continue to 
grow before impurity species block the active sites, whereas if the supersatura- 
tion is raised from a low value, impurities quickly block the sites and stop the 
growth. Several cases of growth rate hysteresis are described by Kubota, 
Yokota and Mullin (1997). 

6.3 Crystal growth and dissolution 

If both crystallization and dissolution processes were purely diffusion con- 
trolled in nature, they should exhibit a true reciprocity; the rate of crystal- 
lization should equal the rate of dissolution at a given temperature and under 
equal concentration driving forces, i.e. at equal displacements away from the 
equilibrium saturation conditions. In addition, all faces of a crystal would 
grow and dissolve at the same rate. These conditions rarely, if ever, occur in 

Crystals usually dissolve much faster than they grow, and up to fivefold 
differences are not uncommon. Different crystallographic faces grow at differ- 
ent rates; they may even dissolve at different rates, but few reliable measure- 
ments of this behaviour have yet been reported. These facts have led most 
investigators to support the view that the crystallization process can be con- 
sidered on the basis of a simple two-step process: bulk diffusion being followed 
by a surface 'reaction' at the growing crystal face (section 6.1.4). There have, 
however, been other suggestions put forward. Some authors have suggested 
that crystals dissolve faster than they grow because the exposed surface is not 
the same in each case; etch pits rapidly form on the faces of a dissolving crystal 
(these occur either at random point defects or points where line defects break 
the surface) as seen in Figure 6.30a. Dissolution then proceeds by a pitting and 
layer-stripping process. It is well known that a broken or etched crystal grows 
initially at a much faster rate than that when the faces are smooth, but as 
Van Hook (1961) has pointed out, even an overgenerous allowance of extra 
surface area due to pitting cannot possibly explain the greater rates of dissolu- 
tion compared with the rate of crystallization of sucrose under comparable 
conditions. Other workers have expressed similar views, and it has been shown 
that some dissolution processes may also involve a slow 'reaction' step at the 
crystal surface (Bovington and Jones, 1970; Zhang and Nancollas, 1991). 

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Crystal growth 261 


, . - 

* <*,' 


Figure 6.30. Growth and dissolution of a sucrose crystal: (a) etch pits appearing at the 
onset of dissolution; (h) growth layers moving over a crystal surface 







"5 20XIO -9 - 










8x!CT 3 I2xl0* 3 I6xl0" 3 20x10"' 

Ac, concentration difference (Kg of hydrate/ kg of solution) 

Figure 6.31. Growth and dissolution for potash alum crystals at 32 °C. Mean crystal 
sizes: ▲ = 1.75, = 1.02, 9 = 0.73, = 0.51, A = 1.69, M = 1.4, = 0.99, + = 0.75, 
V — 0.53 mm. {After Garside and Mullin, 1968) 

Growth and dissolution rates of crystals can be measured conveniently in the 
laboratory fluidized bed crystallizer described above {Figure 6.20). Some typical 
results for potash alum are shown in Figure 6.31, where it can be seen that 
dissolution rates are very much greater than growth rates under equal driving 
forces (Ac). Similar results have been reported for potassium sulphate (Mullin 

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and Gaska, 1969). However, whilst in both cases the dissolution rates are first- 
order with respect to supersaturation, i.e. 

^ D = K D Ac 


the growth processes are not, i.e. Rq = KqAc 8 (equation 6.70) where g ~ 1.6 
for potash alum at 32 °C and g ~ 2 for potassium sulphate at all temperatures 
fom 10 to 50 °C. In equations 6.70 and 6.90 ^d and Kq are the overall 
dissolution and growth mass transfer coefficients, respectively. 

Crystal growth retardants do not necessarily have an influence on the dis- 
solution process, but many such cases have been reported. Sears (1958) showed 
that complex inorganic ions such as FeFg~ can retard both the growth and 
dissolution of lithium fluoride at concentrations of <10~ 5 molL~ . Nancollas 
and Zawacki (1984) commented on the growth and dissolution retardation of 
sparingly soluble salts using, for example, chelating anions that adsorb at 
cationic sites. Kubota et al. (1988) demonstrated that ppm traces of Cr + in 
solution can prevent potassium sulphate crystals dissolving, with the effect that 
solubilities of this salt measured under these conditions are always lower than 
the true equilibrium solubility (section 2.8). 

An example of the effect of trace impurities on both dissolution and growth 
is shown in Figure 6.32 for the case of Fe(III) and a single crystal of potassium 
sulphate (Kubota et al., 1999). The effect of temperature on both growth and 
dissolution processes has been considered in section 6.2.6. 
















" ■ T " - 


-" i 






P* }& 100 ppm" 
/ / (pH=2.33 
pure 10 ppm 




35 40 45 

Temperature [°C] 


Figure 6.32. Dissolution and growth rates (expressed as a mass increase or decrease, 
normalized with the initial seed crystal mass) of a single potassium sulphate crystal in the 
presence o/'Fe(III) as trace impurity added as FeNH^SO^ • 2H2O. (After Kubota et al., 

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Crystal growth 263 

6.3.1 Mass transfer correlations 

Dissolution rate data obtained under forced convection conditions can be 
correlated by means of equation 6.64 or 6.65. As described in section 6.2.2, 
equation 6.64 is the preferred relationship on theoretical grounds, since Sh = 2 
for mass transfer by convection in stagnant solution (Re = 0), whereas equa- 
tion 6.65 incorrectly predicts a zero mass transfer rate (Sh = 0) for this condi- 
tion. However, at reasonably high values of Sh (> 100) the use of the simpler 
equation 6.65 is quite justified. The exponent of the Schmidt number b is 
usually taken to be i and for mass transfer from spheres the exponent of the 
Reynolds number a = \. 

Data plotted in accordance with equation 6.65 for the dissolution of potash 
alum crystals yield the relationship (Garside and Mullin, 1968) 

Sh = 0.37 Re° p 62 Sc 033 (6.107) 

where the particle Reynolds number, Re p , is based on a mean crystal size and its 
relative velocity when suspended in the solution. 

Rowe and Claxton (1965) have shown that heat and mass transfer from a 
single sphere in an assembly of spheres when water is the fluidizing medium can 
be described by 

Sh = A + BRe™Sc xli (6.108) 

where A = 2[1 - (1 - ef s \ B = 2/3e and (2 - 3m)l(3m - \) = 4.65Re; 02S . 
The solution Reynolds number, Re s , is based on the superficial fluid velocity, 
u s , and e = voidage. 

Another correlation used for predicting rates of mass transfer in fixed and 
fluidized beds is that of Chu, Kalil and Wetteroth (1953). The /-factor for 
diffusional mass transfer given by 


Jd=[ — )Sc« i (6.109) 

is plotted against the modified solution Reynolds number Re' s (l — e), where Re' s 
contains L' , the diameter of a sphere with the same surface area as the crystal 
under consideration. The recommended expressions for calculating the mass 
transfer coefficients are: 

1 < Re'J(l -e)< 30: j d = 5.7Re'J(l - e) -0 ' 78 (6.110) 

30 < Re'J(\ -e)< 5000: j d = \.HRe'J(\ - sT ' 44 (6.1 1 1) 

Dissolution rate data for potash alum are plotted in accordance with equations 
6.108 and 6.110 with e = 0.95, in Figure 6.33, where it can be seen that the 
results lie reasonably close (±20%) to the predicted values. However, it should 
be noted that equation 6.110 is very sensitive to values of s as e —> 1, SO it 
cannot be applied with any reliability to very lean beds of dissolving particles 
and certainly not to the dissolution of single particles. 

For the dissolution of crystals smaller than about 60 urn, a rough estimate of 
the diffusional mass transfer rate may be made because as Re p ^ Sh reduces 

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Equation of Rowe and 
Claxton with e = 0.95 

Present results 





1 000 

Equation of Chu et. a/. 
with e =0.95 

S" o.i 


300 1000 3000 

10 000 

Figure 6.33. Comparison of dissolution data for potash alum at 32 °C with the mass 
transfer correlations of Rowe and Claxton and Chu et al. {After Rowe and Claxton, 
1965; Chu, Kalil and Wetteroth, 1953) 

to its limiting value of 2 (equation 6.64), i.e., K& = 2D/L where D is the 
diffusivity (m 2 s _1 ), L the crystal size (m) and Kq the mass transfer coefficient 
for dissolution (ms _1 ). The dissolution time, tu, of fine crystals of size L may 
therefore be expressed as fo = pL 2 /8DAc, where p is the crystal density and 
Ac = c* — c is the undersaturation, the driving force for dissolution, the reverse 
of equation 3.67. 

For crystals larger than about 60 urn in agitated vessels, it is difficult to 
estimate the relative crystal-solution velocity (section 9.4.1), and hence Re p , 
but an order of magnitude estimate of the dissolution mass transfer coefficient 
may be made from the Levins and Glastonbury (1972) equation: 

5/z = 2 + 0.47 



4 3 







where d$ and dy are the diameters of the stirrer and vessel, respectively and e is 
the stirrer energy dissipation rate (Wkg~ ) in the vessel. 

6.3.2 Films and boundary layers 

When a fluid flows past a solid surface there is a thin region near the solid- 
liquid interface where the velocity becomes reduced owing to the influence of 
the surface. This region, called the 'hydrodynamic boundary layer' <5h, may be 

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Crystal growth 265 

partially turbulent or entirely laminar in nature, but in the case of crystals 
suspended in their liquor the latter is most probable. 

For mass transfer processes another boundary layer may be defined, viz. the 
'mass-transfer or diffusion boundary layer', S m . This is a thinner region close to 
the interface across which, in the case of a laminar hydrodynamic boundary 
layer around the crystal, mass transfer proceeds by molecular diffusion. Under 
these conditions the relative magnitudes of the two boundary layers may be 
roughly estimated from 

^«5c'/ 3 (6.113) 

where Sc = rj/p s D is the dimensionless Schmidt number (77 = viscosity, 
p s = solution density, D = diffusivity). 

The ratio of the thicknesses of the two layers depends considerably on the 
solution viscosity and diffusivity. For example, for ammonium alum crystals 
in near-saturated aqueous solution at 25 °C, 77= 1.2 x 10~ 3 kgm~' s , 
D = 4 x 10- 10 m 2 $-\ p s = 1.06 x 10 3 kgirT 3 . Therefore, Sc = 2.8 x 10 3 and 
Sh/S m « 14. However, for sucrose at 25°C, 77= KT 1 , D = 9 x KT 11 and 
ft = 1.5 x 10\ giving Sc = 7.4 x 10 5 and S h /6 m « 90. 

In the description of mass transfer processes another fluid layer is frequently 
postulated, viz. the 'stagnant film' (see Figure 6.8) or, as it is sometimes called, 
the 'effective film for mass transfer', 8. This hypothetical film is not the same 
thing as the more fundamental diffusion boundary layer S m , but it may be 
considered to be of the same order of magnitude. 

The thickness of the effective film for mass transfer, 8, is defined by 

6=^- (6.114) 


where p s = solution density, D = diffusivity and A; is a mass transfer coefficient 
expressed as ms~'. As described earlier, mass transfer data are frequently 
correlated by relationships such as equation 6.65 in which the Sherwood 
number Sh = kL/D and particle Reynolds number Re v = p % uL\r\. L = particle 
size and u = relative particle solution velocity. Exponent b of the Schmidt 
number is generally taken as i and in the case of a laminar boundary layer it 
can be shown theoretically that exponent a of the Reynolds number is 5. 
However, a can vary from about 0.5 to 0.8 if the boundary layer is not truly 
laminar. Values of the constant cf> for granular solids may range from about 0.3 
to 0.9. So, writing a simple, arbitrary form of equation 6.65 as 

Sh = ^RefSc l/3 (6.115) 

and expressing Sh = L/S (using equation 6.1 14), we get 

and this equation has often been used to give a rough estimate of the value of 
S. It should be noted, however, that equation 6.116 depends on the mass transfer 

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266 Crystallization 

process being first-order with respect to the concentration driving force, other- 
wise Sh is not dimensionless and equation 6.1 15 is invalid. A further complica- 
tion (Paterson and Hayhurst, 2000) is that equation 6.114 only strictly applies 
to the case of a planar film of thickness S whereas the appropriate relationship 
for a spherical shell film should be expressed in terms of a characteristic 
distance for mass transfer / where 

l = D/k = L/(2 + L/S) (6.117) 

which leads to the statement that 

Sh = 2 + L/6 = 2+f(Re,Sc) (6.118) 

This equation encompasses two asymptotic results for (i) the stagnant case: 
6/L — ► oo, Sh — > 2 and (ii) the planar case: 6/L — > 0, / — > 6. 

It is thus possible to estimate a value for the thickness of the so-called 
stagnant film, 6, but it is perhaps worthwhile at this point to question the 
meaning and utility of this quantity. The concept of a stagnant film at an 
interface is undoubtedly useful in providing a simple pictorial representation 
of the mass transfer process, but in the case of crystals growing or dissolving in 
multi-particle suspensions the actual existence of stable films, of the magnitude 
normally calculated as shown above, around each small particle is debatable, to 
say the least. Further, the value of 6 can only be deduced indirectly from the 
mass transfer coefficient and diffusivity (equation 6.114), and it is difficult to 
select the appropriate value of D to use in any given situation. The question 
arises, therefore, as to whether or not S is a meaningful quantity to calculate 
in these circumstances. In any case, the hypothetical nature of the stagnant 
film should be clearly appreciated, and calculated values of its thickness should 
be used with considerable caution. 

6.3.3 Driving forces for mass transfer 

There is a wide choice of possible driving forces for a mass transfer process, but 
provided that the driving force is clearly defined the selection is generally of 
little importance. However, in certain cases, e.g. under conditions of high mass 
flux, the choice becomes critical. 

For low mass flux mass transfer from a single sphere to an extensive fluid, the 
general correlation 

5// = 2 + 0J2Re l J 2 Sc l1 ' 3 (6.1 19) 

may be used over the range 20 < Re p < 2000. 

The mass transfer coefficient in the Sherwood number may be defined by 

R = k e (c -c oo ) (6.120) 

= k c (p U>Q ~ PooWoo) (6.121) 

= p s k c (u> - Woe) (6.122) 

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Crystal growth 267 

since for low mass flux po ~ p x ~ p s . Other definitions of the mass transfer 
coefficient include 

R = k u (wo-w< x >) (6-123) 

= k y (Y -Y oo ) (6.124) 

= k b B (6.125) 

In equations 6.120-6.125 R = mass flux (kgm s _1 ), c = solution con- 
centration (kgm~ ), k = mass transfer coefficient (fc c = ms -1 , k u = 
kgm~ 2 r'Aw" 1 , k y = kgm~ s~'Ay~' and k\> = kgm~ s~ l B~ l ), p s = solution 
density (kgm~ 3 ), u> = mass fraction of solute in solution (dimensionless) and Y 
is the mass ratio of solute to solvent in the solution (dimensionless). The 
subscripts and oo refer to the interfacial and bulk solution conditions, 

The dimensionless mass transfer driving force B is defined by 

B= ^-^ (6.126) 

w t - ^o 

where u t is the mass fraction of the solute in the transferred solid substance, i.e. 
u> t = 1 for a single component. If the solute is a hydrate, then w t = 1 only if the 
mass fractions are expressed as mass of hydrate per unit mass of solution. 

Equation 6.1 19 should describe the dissolution of a solid solute into a solvent 
or its own solution, and either k c or k w can be used, as Sh = k c d/D = k^d/p^D. 
However, complications can arise if the solute solubility is high. First, the 
concentration dependence of the physical properties become significant and, 
since po ^ Poo, the Sherwood numbers based on k c and k^ will not be equal. 
Second, the mass flux from the surface of the solid alters the concentration 
gradient at the surface compared with that obtained under otherwise identical 
conditions of low mass flux. 

Diffusion coefficients of electrolytes in water are greatly dependent on con- 
centration; variations of ±100% from infinite dilution to near-saturation are 
not uncommon. Moreover the change is often non-linear and accurate predic- 
tion of its effect is extremely difficult. Other physical properties, such as 
viscosity and density, change over this concentration range but not to such 
an extent. 

The effects of concentration dependent physical properties on the correlation 
of dissolution mass transfer data have been reported in some detail by Nienow, 
Unahabhoka and Mullin (1966, 1968). 'Mean' solution properties should be 
used for the Sherwood and Schmidt groups in equation 6.119 if the mass 
transfer data for moderately soluble substances are to be correlated effectively. 
The arithmetic mean will suffice for viscosity and density, but the integral value 
must be used for the diffusivity (equation 2.27). Bulk solution properties are 
used for the Reynolds number. 

For low to moderate mass flux mass transfer studies, therefore, provided that 
the physical property changes are taken into account, mass transfer coefficients 
k c or k^ may be used. The dimensionless mass ratio driving force, AY, has been 
used quite successfully in crystallization and dissolution studies (Garside and 

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268 Crystallization 

Mullin, 1968; Mullin and Gaska, 1969), but this has the disadvantage that each 
value of Woo yields a different value of 0, even if the physical property variations 
are allowed for. 

However, if the dimensionless driving force, B, is used, together with the 
appropriate physical properties, the value of <f> in equations 6.64 and 6.65 
remains substantially constant at about 0.7-0.8 for a wide range of systems. 
There is little doubt that B is the best driving force to use for high mass flux 

A comprehensive account of the role of transport processes in crystallization 
has been given by Garside (1991). 

6.3.4 Mass transfer in agitated vessels 

Crystallization and dissolution data obtained from agitated vessel studies may 
be analysed by the methods discussed above, but a survey of the literature 
related to the subject of solid-liquid mass transfer in agitated vessels shows that 
there is an extremely wide divergence of results, correlations and theories. The 
difficulty is the extremely large number of variables that can affect transfer 
rates, the physical properties and geometry of the system and the complex 
liquid-solid-agitator interactions. 

Relationships such as equations 6.64 and 6.65 are commonly used for correl- 
ating solid-liquid mass transfer data. However, the Reynolds number should 
not be based on the agitator dimensions and speed, because this cannot take 
into account one of the most important factors, viz. the state of particle 
suspension. The mass transfer coefficient increases sharply with agitator speed 
until the particles become fully suspended in the liquid, after which the rate of 
increase with further increases in speed is reduced considerably. A maximum 
rate of mass transfer occurs when substantial aeration of the liquid occurs at 
high agitator speeds. From the 'just-suspended' to 'severe aeration' conditions 
the mass transfer coefficient may be enhanced by 40-50% while the agitator 
power input may be increased tenfold. There is little justification, therefore, for 
using agitator speeds much higher than those needed to suspend the particles in 
the system. 

The appropriate velocity term for the particle Reynolds number in equations 
6.64 and 6.65 is the slip velocity, i.e. the relative velocity between particle and 
fluid. The slip velocity is usually assumed to be the free fall velocity of the 
particle, but this quantity is not easy to predict. 

The critical mass transfer rate, for particles just suspended in a liquid, can be 
estimated from equation 6.119, the 'mean' solution properties being used as 
explained above. The terminal velocity, U\, for use in the Reynolds number may 
be calculated from the empirical equations 

u t = 0.153 g 071 L L14 Ap - 7 V- 2 V°' 43 (6.127) 

for particles smaller than 500 um, and from 

ut = (4gLAp/3 Ps f 2 (6.128) 

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Crystal growth 269 

for particles larger than 1500|im (Nienow, 1969). For particles of intermediate 
size, u t should be predicted from both relationships, and the smaller value used 
in the Re p as a conservative estimate. In equations 6.127 and 6.128, u t = cm s , 
g = 981 cms -2 , L = cm, p = gcm~ 3 and r\ = poise (g$ cm ) where p s and r\ 
refer to the bulk solution. Ap is the solid-liquid density difference. 

The expected mass transfer coefficient can be predicted from the critical 
value by multiplying by an enhancement factor ranging from about 1.1 for 
particles ~200 |J.m to about 1.4 for particles ~5mm. Particle density also 
influences the rate of mass transfer. The reason for this enhancement is the 
increased level of turbulence at which larger and denser particles become 
suspended in the liquid. 

Another model for mass transfer is based on the Kolmogoroff theory of 
homogeneous isotropic turbulence adapted to solid-liquid systems (Kolaf, 
1958, 1959; Middleman, 1965; Hughmark, 1969). The energy put into the 
system by the agitator is considered to be transferred first to large-scale eddies 
and then to larger numbers of smaller isotropic eddies from which it is dis- 
sipated by viscous forces in the form of heat. For a given system the mass 
transfer coefficient, k, can be related to the energy input, e, to the system by 
/coce 025 . 

The Kolmogoroff theory can account for the increase in mass transfer rate 
with increasing system turbulence and power input, but it does not take into 
consideration the important effects of the system physical properties. The 
weakness of the slip velocity theory is the fact that the relationship between 
terminal velocity and the actual slip velocity in a turbulent system is really 
unknown. Nevertheless, on balance, the slip velocity theory appears to be the 
more successful for solid-liquid mass transfer in agitated vessels. 

6.4 Crystal habit modification 

6.4.1 Crystal morphology and structure 

The morphology of a crystal depends on the growth rates of the different 
crystallographic faces. Some faces grow very fast and have little or no effect 
on the growth form; the ones that have most influence are the slow-growing 
faces. The growth of a given face is governed by the crystal structure and 
defects on the one hand, and by the environmental conditions on the other. 

A number of attempts have been made to predict the equilibrium form of 
a crystal. According to the Bravais rule (chapter 1), the important faces govern- 
ing the crystal morphology are those with the highest reticular densities and the 
greatest interplanar distances, dhki- Or, in simpler terms, the slowest-growing 
and most influential faces are the closest-packed and have the lowest Miller 
indices. The surface energy theories of crystal growth (section 6.1.1) suggest 
that the equilibrium form should be such that the crystal has a minimum total 
surface free energy per unit volume. 

The morphological theory of Hartman and Perdok (1955) considers the bond 
energies involved in the integration of growth units into the lattice. In this 

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theory crystal growth is considered to be controlled by the formation of strong 
bonds between crystallizing particles. A strong bond is defined as a bond in the 
first co-ordination sphere of a particle. The two-dimensional crystal shown in 
Figure 6.34a is bounded by straight edges that are parallel to uninterrupted 
chains of strong bonds. Such a straight edge is formed when the probability of 
a particle being integrated is greater for site A than site B. In the case illustrated 
the particle at site A is bonded to the crystal with one strong bond more than at 
site B. The uninterrupted chains of strong bonds have been called periodic bond 
chains (PBC); and as the number of strong bonds per unit cell is limited, there 
exists a maximum length for the period of a PBC and, hence, a limited number 

The minimum thickness of a growth layer is the elementary 'slice', dhkl, and 
faces that grow slice after slice are called flat or F-faces. The condition for 
a slice to exist is that two neighbouring parallel periodic bond chains be bonded 
together with strong bonds {Figure 6.34b). If this is not so, no slice exists, i.e. no 
layer growth can occur. Such faces are called stepped or S-faces (Figure 6.34c). 

If no PBC exists within a layer, dhkl, the face is called a kinked or K-face, 
which needs no nucleation for growth since it corresponds to a generalized type 




Figure 6.34. (a) Two-dimensional crystal. Each circle represents a growth unit of KosseFs 
repeatable step, (b) and (c) Projection of a three-dimensional crystal along a PBC. Each 
circle represents a PBC. An F-face results when neighbouring PBCs are linked together by 
strong bonds, otherwise an S-face develops 

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Crystal growth 


Figure 6.35. Crystal with three PBCs parallel to [100] (A), [010] (B), and [001] (Q. The 
F-faces are (100), (010) and (001). The S-faces are (110), (101) and (Oil). The K-face is 
(1 1 1). (After Hartman, 1963) 

of Kossel's repeatable step (Figure 6.35). In terms of crystal structure depend- 
ent growth, therefore, the growth form should be bounded by F-faces only, 
although not all F-faces need be present. 

The Hartman-Perdok approach is applied by making projections of the 
crystal structure parallel to a PBC and tabulating all the bonds. The packing 
of the chains determines the F-faces, provided that the chains are bonded by 
strong bonds. Sometimes it is easier to recognize the slices, and in that case the 
PBC may be found as the intersection of two slices. 

Some reported examples of the use of PBC analysis to predict crystal 
morphology include: hexamethylenetetramine (Hartman and Perdok, 1955), 
calcium sulphate (gypsum) (van Rosmalen, Marchee and Bennema, 1976), 
anthracene (Hartman, 1980), magnesium hydrogenphosphate (newberyite) 
(Boistelle and Abbona, 1981), sodium sulphite and potassium sulphate (Follner 
and Schwarz, 1982), succinic acid (Davey, Mullin and Whiting, 1982), sucrose 
(Aquilano et al., 1983). 

Docherty and Roberts (1988) developed an alternative technique which 
included the computation of surface attachment energies: faces with the lowest 
attachment energies will be the slowest growing and hence the most dominant 
morphologically (Bennema and Hartman, 1980). This approach led to the 
successful modelling of the theoretical morphologies of molecular crystals, 
e.g. anthracene, biphenyl and /3-succinic acid. In a similar manner, Clydesdale 
and Roberts (1991) predicted the structural stability and morphologies of 
crystalline C18-C28 «-alkanes. Anwar and Boateng (1998) have shown how 
crystallization from solution can be simulated using the method of molecular 
dynamics for a model solute/solvent system consisting of atomic species char- 
acterized by the Leonard-Jones potential function. Accounts of molecular 
modelling techniques, based on computer simulation and computational chem- 
istry, are given by Docherty and Meenan (1999) and Myerson (1999). 

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272 Crystallization 

6.4.2 Interface structure 

Consideration of the structure of a growing crystal face can provide additional 
information to assist in the task of crystal morphology prediction. The surface 
roughness (on the molecular level) quantified by the a-factor (section 6.1.7) is 
governed by energetic factors arising from fluid-solid interactions at the inter- 
face between the crystal and its growth environment. The degree of roughness 
of a given crystal face can have an important bearing on the growth mechanism 
controlling its development. A significant change in the a-factor could con- 
siderably alter the face growth potential and hence affect the overall crystal 

A change of solvent often changes the crystal habit and this may sometimes 
be explained in terms of interface structure changes. In general, the higher the 
solubility of the solute in the solvent, the lower the a-value and hence 
the rougher the surface. A smooth face (high a-value) would favour growth 
by the BCF screw-dislocation mechanism, a rough face (low a-value) would 
favour diffusion-controlled growth, while a face of intermediate roughness 
would tend to grow by the B + S mechanism. Since these three mechanisms 
imply different v— a relationships (section 6.1.7), the face growth rates could be 
quite different in different solvents, and any differences in the relative rates of 
growth would manifest themselves in a habit change. 

A detailed study on solvent effects relating to the growth of succinic acid 
crystals from water and isopropanol solutions was reported by Davey, Mullin 
and Whiting (1982). The faster growth of the (010) and (001) faces in water than 
in isopropanol resulted in a succinic acid habit modification from platelets to 
needles, as shown in Figure 6.36. Calculated a-factors for the two faces were 
found to be similar for both solvents, so the change of habit was considered to 
result from chemical interaction with the solvent. Succinic acid interacts, pre- 

3=1 c x 

i° N 


[°E 1 






Figure 6.36. Habits of succinic acid crystals grown from (a) water (b) isopropanol. 
{After Davey, Whiting and Mullin, 1982) 

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Crystal growth 


sumably through hydrogen bonding, more strongly with isopropanol than with 
water, and the stronger adsorption would reduce the face growth rates below 
those in aqueous solution. On the (001) faces, the carboxylic acid groups are 
normal to the surface and adsorption would reduce surface diffusional flux to 
the growth steps. On the (010) faces, the carboxylic acid groups are parallel to 
the surface and adsorption would be active in blocking kink sites. This appears, 
therefore, to be a case in which adsorption effects dominate the growth kinetics. 
In another attempt to explain the habit changes of succinic acid in water/ 
isopropanol solvents, van der Voort (1991) assumed that solvent interactions 
determine diffusion rates. 

The adsorption of an impurity on a crystal face can have a similar effect to 
a change of solvent. Since adsorption reduces the interfacial tension, it will also 
reduce the a-factor and consequently roughen the surface. If adsorption is 
selective, i.e. only on to specific faces of the crystal, or to different extents on 
different faces, any significant change from the smooth to rough condition 
could lead to faster growth on those faces and hence to a habit change. 

Crystal growth enhancement by the adsorption of a foreign species appears 
to be contrary to the commonly held view of the action of an additive in which 
foreign species adsorb at various sites on a crystal face, impede the flow of 
growth layers and reduce the growth rate (section 6.2.8). However, the two 
effects can sometimes be seen in the same system, with growth enhancement 
occurring at low impurity levels followed by a reversal at higher levels when the 
blocking effect becomes dominant. An example is shown in Figure 6.37 where 
the cube (100) and octahedral (111) faces of lead nitrate growing in the presence 
of increasing amounts of methylene blue (Bliznakov, 1965) both exhibit 
a reversal effect at the low impurity level of approximately 5 mg L~ ' . Similar 
examples have been reported with other systems by Budz, Jones and Mullin 
(1986) and Eidelman, Azoury and Sarig (1986). 

The chemisorption of an impurity can cause chemical changes in the crystal 
surface that give it a new structural appearance. The growth of octahedral 

20 40 

Concentration of methylene blue (mg L" 

Figure 6.37. The influence of methylene blue on the (100) and (III) face growth rates of 
lead nitrate at 25 °C and S = 1.08, showing a reversal of effect. (After Bliznakov, 1965) 

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274 Crystallization 

sodium chloride crystals from solutions containing cadmium chloride results 
not from the simple adsorption of Cd 2+ , but from the formation of a new 
phase, CdCi2 • 2NaCl • 3H2O, which has an epitaxial fit with the {111} planes 
of NaCl causing the (111) growth rate to decrease and hence dominate the habit 
(Boistelle and Simon, 1974). 

6.4.3 Structural compatibility 

It is usually assumed that there is some form of affinity between an active 
impurity and the crystallizing species, and this can take a large variety of forms. 
For example, there may be some degree of structural compatibility between 
ionic groups in the modifying agent and the crystal, e.g. as in the case of 
calcium carbonate being modified by metaphosphates, or nitrilotriacetamide 
and nitrilotripropionamide for modifying NaCl and KC1, respectively (Sarig, 
Glasner and Epstein, 1975). There may be some structural similarity with the 
crystal to be modified, particularly in organic systems, and this has led to the 
use of the term 'tailored' crystal growth. A tailored additive usually has two 
parts, one which is structurally compatible with a grouping on one of the 
crystal faces and the other which acts as a repellent, i.e., after integration it 
will then disrupt the subsequent bonding sequence and hence retard the growth 
process on that face. 

A simple example of a tailor-made habit modifier for benzamide was 
reported by Berkovitch-Yellin et al. (1982). This substance normally crystallizes 
from ethanol solution in the form of platelets, with the slowest growth in the 
c direction. During growth the benzamide molecules develop a ribbon pattern 
in which hydrogen bonded cyclic dimers are interlinked by N — H- • O bonds 
along the b axis. The ribbons are stacked along the a axis to yield (001) layers. 
Three different impurities, benzoic acid, o-toluamide and />-toluamide, which 
all bear a structural resemblance to benzamide but contain substituent groups 
that interfere with the bonding, were found to be capable of retarding the 
growth rates along the b, a and c axes, respectively. 

Figure 6.38 demonstrates the action of benzoic acid which, after substituting 
for a benzamide molecule in the lattice by H bonding, repels the next incoming 
benzamide molecule as it encounters an O- • O repulsion due to the lone pair 
electrons of the benzoic acid carbonyl oxygen. The rate of growth along the 
c axis is thus impeded. 

A tailored modifier does not always have to be deliberately added to a cryst- 
allizing system; it may already exist, e.g., as a synthesis by-product of a chemical 
reaction. If its presence causes a crystal habit problems, it must be removed or 
deactivated. On the other hand, it may have a beneficial effect. These are both 
commonly encountered cases in the manufacture of organic chemicals. 
A simple, but industrially important, example is the effect of biuret on the 
crystallization of urea (Davey, Fila and Garside, 1986). In the synthesis of urea 
(NH2CONH2) from ammonia and carbon dioxide a small amount of biuret 
(NH2CONHCONH2), a condensation dimer, is formed. The presence of biuret 
is actually beneficial because from pure aqueous solution urea crystals form as 
elongated [001] needles that are difficult to process. Biuret retards growth in the 

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Crystal growth 275 








— H — N 






€K )~© 



0--H — N 

Benzoic acid 


growth direction 

N ' X N-H 

/ repulsion 

In- coming 
benzamide molecule 


Figure 6.38. Benzoic acid, as an impurity, retarding the growth of benzamide. (After 
Berkovitch-Yellin et al., 1982) 

[001] direction resulting in short stubby urea crystals which are more easily 
handled in the subsequent downstream processes of filtration, washing and 
drying (section 9.7). 

The needle-like morphology of urea results from the strong intermolecular 
hydrogen bonding along the urea crystal c-axis, as shown by the dotted lines in 
Figure 6.39. The urea structure is such that the {001} surfaces cannot easily 
discriminate between two urea molecules and one biuret molecule, so biuret 
molecules can easily become attached to the lattice at growth sites in the [001] 
direction. However, subsequent urea molecules attempting to attach to 
a biuret-contaminated surface meet a resistance since the NH2 groups in the 
crystal surface that are needed to form hydrogen bonds are now missing. The 
growth rate in the [001] direction is thus effectively reduced, and stubby crystals 
are the result. This example illustrates the general rule that the most effective 
habit modifiers are those that are able to enter the growing surface and yet once 
there they disrupt further growth. To perform this function effectively the 
additive molecule must resemble the crystallizing molecule while containing 
some small difference in stereochemistry or functionality, rendering it capable 
of inhibiting growth in a selected direction (Davey and Garside, 2000). 

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Figure 6.39. Showing a biuret molecule occupying two urea sites on the fast growing (001) 
face and impeding further growth. {After Davey and Carside, 2000). 

Authoritative account of the control of crystal morphology by the use of 
tailor-made additives have been given by Davey, Polywka and Maginn (1991), 
Popovitz-Biro et al. (1991), Myerson (1999) and Davey and Garside (2000). 

6.4.4 Industrial importance 

Most habit modification cases reported in the literature have been concerned 
with laboratory investigations, but the phenomenon is of the utmost import- 
ance in industrial crystallization and by no means a mere laboratory curiosity. 
Certain crystal habits are disliked in commercial crystals because they give the 
crystalline mass a poor appearance; others make the product prone to caking 
(section 7.6), induce poor flow characteristics or give rise to difficulties in the 
handling or packaging of the material. For most commercial purposes a granu- 
lar or prismatic habit is usually desired, but there are specific occasions when 
other morphologies, such as plates or needles, may be wanted. 

In nearly every industrial crystallization some form of habit modification 
procedure is necessary to control the type of crystal produced. This may be done 
by controlling the rate of crystallization, e.g. the rate of cooling or evaporation, 
the degree of supersaturation or the temperature, by choosing a particular 
solvent, adjusting the solution pH, deliberately adding an impurity that acts as 
a habit modifier, or even removing or deactivating some impurity that already 
exists in the solution. A combination of several of the above methods may have 
to be used in specific cases, as seen in the examples quoted in section 6.2.8. 

Many dyestuffs act as powerful habit modifiers for inorganic salts. Buckley 
(1952) has summarized a large number of case histories giving an indication of 
the concentrations necessary to induce the required change, but these additives 
do not nowadays find any significant industrial application. 

Surface-active agents (surfactants) are frequently used to change crystal 
habits. Common anionic surfactants include the alkyl sulphates, alkane sulphon- 

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Crystal growth 


ates and aryl alkyl sulphonates. Quaternary ammonium salts are frequently 
used as cationic agents. Non-ionic surfactants only occasionally find applica- 
tion as habit modifiers. 

Polymeric substances such as polyvinylalcohol, polyacrylates, polyglutam- 
ates, polystyrene sulphonates, alginates, polyacrylamides, etc., have also found 
application, as have long-chain and proteinaceous materials like sodium car- 
boxymethylcellulose, gelatin and phosphoproteins. Sodium triphosphate, 
sodium pyrophosphate, organic derivatives of phosphonic acid (H3PO3), low 
molecular weight organic acids, such as citric, succinic and tartaric and their 
derivatives, are also useful habit modifiers. 

The trace presence of foreign cations can exert an influence on the crystal 
habit of inorganic salts. Some act by simple substitution in the lattice, e.g. Cd 2+ 
for Ca 2+ in calcium salts or Ca 2+ for Mg 2+ in magnesium salts, as a result of 
similar ionic radii and charge. Trivalent cations, particularly Cr 3+ and Fe 3+ , 
have a powerful effect on the morphology of salts such as ammonium and 
potassium dihydrogenphosphates (Mullin, Amatavivadhana and Chakraborty, 
1970; Davey and Mullin, 1974, 1976) and ammonium sulphate (Larson and 
Mullin, 1973). These trivalent cation habit modifiers are not only powerful in 
effect, i.e., active at very low concentrations in the system, but also that above 
some critical concentration they begin to have a severe disruptive effect on the 
overall crystal growth process, resulting in the production of unacceptable 
crystalline products. For example, at a Cr 3+ concentration of 5 ppm the normal 
orthorhombic crystal habit of ammonium sulphate changes with the appear- 
ance of higher index faces, while at around 20 ppm large grotesque non-faceted 
crystals are produced (Figure 6.40). 

Complex cations, like Fe(CN) 6 ~, have a remarkable influence on sodium 
chloride (Figure 6.41). At concentrations of around 0.1%, excrescences develop 
at the corners of the normal cubic crystals producing large hard crystals with a 
skeletal appearance, often referred to as dendrites (see Figure 8.3). At around 
1% Fe(CN) 6 ~, however, the product changes to soft friable particles with little 
or no outward appearance of crystallinity (Cooke, 1966). 

Phoenix (1966) has reported on the effects of a wide variety of inorganic and 
organic additives on NaCl, NaBr, KC1, KCN, K 2 S0 4 , NH 4 C1, NH4NO3 and 
(NH4) 2 S04. A considerable amount of valuable quantitative information is 
given concerning the effects of the different additives on crystal habit, growth 
and dissolution rates, and anti-caking effectiveness. The influence of ferrocyan- 
ide ions in producing dendritic crystals of NaCl is discussed in some detail. 

Figure 6.40. Habit changes in ammonium sulphate crystals caused by traces of impurity: 
(a) pure solution, (b) 5 ppm Cr 3+ , (c) 20 ppm Cr 3+ . (Larson and Mullin, 1973) 

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278 Crystallization 


Figure 6.41. Habit changes in sodium chloride crystals caused by traces of impurity: 
(a) pure solution, (b) 0.1% Fe(CN)^, (c) 1% Fe(CN)^. (Cooke, 1966) 

There are literally thousands of reports in the scientific literature concerning 
the effects of impurities on the growth of specific crystals, and it would be 
superfluous to attempt a summary here. General reviews on the influence of 
additives in the control of crystal morphology have been made by Kern (1965), 
Boistelle (1976), Davey (1979), Botsaris (1982), Nancollas and Zawacki (1984), 
van Rosmalen, Witkamp and de Vreugd (1989), Davey et al. (1991) and Pfefer 
and Boistelle (1996). 

The selection of a suitable habit modifier for the industrial production of 
crystals of a particular form normally begins with a series of laboratory-scale 
screening tests covering a wide range of potential additives at different concen- 
trations. It is usually necessary to conduct further trials with the more promising 
modifiers to attempt to identify the ones that should prove efficacious on the 
industrial scale. Quite clearly, all these procedures can be extremely time-con- 
suming and costly, but ultimate success depends on the key step of deciding 
which additive is likely to be potentially useful. There is a rapidly growing 
interest, therefore, in an alternative procedure to eliminate guesswork and 
serendipity from the initial selection process, involving the use of computer 
modelling techniques to match additive molecular species with the molecular 
configurations on the specific faces of the crystal that need to be influenced. This 
is a rapidly developing field of activity, but it should be understood that whilst 
the molecular modelling approach to habit modification undoubtedly holds 
great promise for the future, it is first necessary to be in possession of detailed 
crystallographic data and quantifications of intermolecular bond strengths at 
relevant crystal faces. Unfortunately, this information is not always readily 
available. Nevertheless, there are already several reported examples of the 
successful application of the molecular modelling approach to habit modifica- 
tion (Davey, Polywka and Maginn, 1991; Lewtas et al., 1991; Lee et al., 1996; 
Myerson, 1999; Davey and Garside, 2000; Winn and Doherty, 2000). 

Maximum crystal size 

Theoretically there is no limit to a product crystal size, but there is generally 
a practical limit. It is common experience that some crystals do not normally 
grow beyond a certain size in agitated industrial crystallizers (Figure 6.42), 
although there is no single clear-cut answer to this problem. 

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Crystal growth 



I0 4 

I0 3 

?. io z 






10- I0 U 10' 

Relative supersaturation, o- 

I0 3 

Figure 6.42. Maximum mean crystal sizes obtained in an MSMPR crystallizer: A, KC1; 
B, NaCl; C, (NH 2 ) 2 CS; D, (NH 4 ) 2 S0 4 ; E, KN0 3 ; F, Na 2 S0 4 ; G, K 2 S0 4 ; H, 
NH 4 A1(S0 4 ) 2 ; /, K 2 Cr 2 7 ; /, KA1(S0 4 ) 2 ; K, KCIO3; L, NiS0 4 (NH 4 ) 2 S0 4 ; M, BaF 2 ; 
N, CaCO,; O, Ti0 2 ; P, CaF 2 ; Q, BaS0 4 . {After Mersmann and Kind, 1988) 

Some crystals have such low growth rates that excessive residence times 
would be necessary to produce large crystals. For example, at a linear growth 
rate of 10~ 7 m s _1 a nucleus would grow to 1 mm in just over 1 h, but at 10 -9 m s _1 
it would require around 6 days. Growth rates exhibited by inorganic salts in 
aqueous solution generally lie well within this range (Table 6.1). Of course, 
increased residence time alone in an agitated crystallizer may not greatly 
influence the product size because of the inevitable occurrence of secondary 
nucleation (section 9.1.1) which greatly increases the number of product 
crystals and consequently inhibits the development of large crystals. Growth 
rates can be increased by raising the operating level of supersaturation, but 
nucleation rates are even more sensitive to supersaturation and play the 
dominant role. 

The presence of impurities in the system can also have a significant effect. 
For example, crystallization of copper and cadmium sulphates from plating- 
bath liquors, to which gelatin has been added, produces crystals no larger than 
1 urn, yet both of these salts can readily be crystallized from normal aqueous 
solution as large crystals (> 1 mm). 

Some crystals appear to become prone to attrition once they have been 
grown beyond a certain critical size in an agitated crystallizer. To some extent 
this can be attributed to increased damage from the agitator as higher rota- 
tional speeds are needed to keep them in suspension. Sometimes the critical size 
coincides with the onset of polycrystalline growth which tends to make the 
crystals friable. Polycrystalline growth, however, may not only render the 
crystals mechanically weak, but may even make the crystals fhermodynamically 

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280 Crystallization 

unstable (the Gibbs-Thomson effect - section 3.7) and dissolution would tend 
to occur at sharp edges and grain boundaries, i.e. at regions of very small 
radius, and cause the crystal to achieve a rounded shape. It is possible, there- 
fore, that opaque egg-shaped crystals produced in many industrial crystallizers, 
are as much the result of sequences of crystallization-dissolution as of attrition. 

6.5 Polymorphs and phase transformations 

It is not uncommon in crystallization processes for the first crystalline phase to 
make its appearance to be metastable, e.g. a polymorph or hydrate (Ostwald's 
rule of stages - section 5.7). Some metastable phases rapidly transform to a more 
stable phase while others can exhibit apparent stability for an exceptionally 
long time. Some transformations are reversible (enantiotropic) while others are 
irreversible (monotropic), as explained in sections 1.8 and 4.2.1. In some cases, 
the metastable phase may have more desirable properties than the stable phase, 
e.g., a metastable pharmaceutical product may be more pharmacologically 
active than the stable form. If the required metastable form is first to crystallize, 
it is important to isolate and dry it quickly to prevent it transforming to the 
stable form. Once in the dry condition a metastable form can often remain 
unchanged indefinitely. If the stable polymorph is required, it is essential to 
create conditions and allow sufficient time in the crystallizer for total trans- 
formation to the more stable phase to be ensured. 

Polymorphism is commonly encountered in crystalline substances. Calcium 
carbonate, for example, has three polymorphs, ammonium nitrate has five 
(section 1.8), and some organic compounds have many more. Aspirin, for 
example, was once thought to have 6 or 8 and phenobarbitone as many as 
13, but it is always worth keeping in mind the somewhat provocative comment, 
generally attributed to McCrone (Dunitz and Bernstein, 1995), that the number 
of polymorphs discovered often seems to be proportional to the time and 
money spent looking for them. 

Because polymorphs differ in the type of lattice, or in the spacing of the 
lattice points, they can exhibit different crystalline shapes and may often be 
readily identified by visual or microscopic observation. These characteristics, 
however, should not be confused with changes in crystal habit (section 6.4) 
which are caused solely by changes in the relative rates of growth of specific 
faces and do not affect the basic physical properties of the substance. 

All crystals of one given substance, which may exhibit different habits, have 
identical physical properties. On the other hand, the different polymorphs of a 
given substance, which may also differ in habit, will exhibit different physical 
properties: density, hardness, melting point, solubility, reactivity, thermal prop- 
erties, optical and electrical behaviour, etc. Each polymorph constitutes a 
separate phase of the given substance, in the Gibbs' phase rule sense, whereas 
crystals of different habit constitute the same phase. Polymorphs may trans- 
form in the solid state, but crystals of different habit cannot. 

Strictly speaking, hydrates and other solvates are not polymorphs because 
they are different chemically from their parent compounds, although they do 

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Crystal growth 281 

have some similar characteristics to polymorphs such as being capable of 
transformation to more stable forms. Similarly, enantiomorphs (section 1.9) 
are not true polymorphs, although they share many of their characteristics, e.g., 
they do have different lattice structures, yet they are chemically identical. 
Furthermore, unlike polymorphs they have identical solubilities and/or melting 
points and cannot transform into one another. The separation of mixtures of 
enantiomers is discussed in section 7.1.3. Bernstein, Davey and Henk (1999) 
have drawn attention to a little appreciated phenomenon of the simultaneous 
crystallization of different polymorphs. They use the term 'concomitant poly- 
morphs' to describe these mixtures that can occur and cause problems in 
industrial processes since the product crystals may show erratic variations in 
habit, colour, melting point, dissolution rates, etc., despite any evidence of 
process changes or impurity contamination. 

Polymorphs and solvates can be identified and characterized by several 
analytical techniques including powder X-ray diffraction, IR and NMR spec- 
troscopy. Differential scanning calorimetry (DSC) is useful for monitoring 
phase transformations and the hot-stage microscope is best for the identifica- 
tion of concomitant polymorphs. 

Under specified conditions of temperature and pressure, except at a trans- 
ition point, only one polymorph is thermodynamically stable. All others are 
unstable and potentially capable of transforming to the stable polymorph. 
Whether they will do so, however, is another matter entirely. The more stable 
polymorph has the lower free energy at a given temperature. If polymorph II is 
more stable than polymorph I then the chemical potential of the species in the 
solid phase II is lower than that in solid phase I, i.e. 

m<m (6-129) 

Under equilibrium conditions, i.e. for the solid phase in contact with its 
saturated solution, the chemical potentials are identical for each species in the 
solid and solution phases, so it is possible to write 

fi Q + RT\nan < (J-o + 'RTlnai (6.130) 

where liq is the standard chemical potential and a is the solution activity, both 
being expressed on a common basis (section 3.6.1). Therefore, 

an < a\ (6.131) 

and, since activity a and concentration c are related, 

en < ci (6.132) 

which leads to the important statement that, at a given temperature, the more 
stable phase will always have the lower solubility in any given solvent. Similarly, 
at a given pressure, the more stable phase will always have the higher melting 
point, but this information cannot be regarded as an infallible guide to the 
relative stability at some other temperature well below the melting point. 

Typical solubility diagrams for substances exhibiting monotropic and enan- 
tiotropic behaviour are shown in Figure 6.43. In Figure 6.43a, form II, having 

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— re 



(U.)/ > 




(1 J-U 

* — I 







Figure 6.43. Solubility curves exhibiting (a) monotropy, (b) enantiotropy, (c) enantiotropy 
with metastable phases 

the lower solubility, is more stable than form I. These two non-interchangeable 
polymorphs are monotropic over the whole temperature range depicted. 

In Figure 6.43b, form II is stable at temperatures below the transition 
temperature T and form I is stable above T. At the transition temperature both 
forms have the same solubility and reversible transformation between these two 
enantiotropic forms I and II can be effected by temperature manipulation. 
Figure 6.43c, however, depicts the intervention of metastable phases (the 
broken line extensions to the two solubility curves) which bear evidence of the 
importance of kinetic factors which for a time may override thermodynamic 
considerations. For example, if a solution of composition and temperature 
represented by point A (supersaturated with respect to both I and II) is allowed 
to crystallize it would not be unusual if the metastable form I crystallized 
out first even though the temperature would suggest that form II is the stable 
form. This would simply be an example of Ostwald's rule (section 5.7) being 
followed. This behaviour would occur, for example, if form II had the faster 
nucleation and/or crystal growth rates. However, if the crystals of form I were 
kept in contact with the mother liquor, transformation could occur as the more 
soluble form I crystals dissolve and the less soluble form II crystals nucleate 
and grow. 

An example of solvent-mediated transformation may be seen in Figure 6.44 
which comprises six frames from a time-lapse cine-micrograph showing meta- 
stable anhydrous sodium sulphate dissolving while the stable phase 
Na2SC>4 • IOH2O nucleates and grows. Figure 6.45 shows two stages in the 
transformation of metastable vaterite into stable phase calcite crystals during 
the precipitation of calcium carbonate from aqueous solution. 

Transformation is not certain even though a system enters a condition that 
will theoretically allow it. Transformation can only be ensured if a more stable 
solid phase is already present, is introduced, e.g. by deliberate seeding, or 
makes its appearance by nucleation. The rate of transformation may be influ- 
enced by retarding the rate of dissolution of the less stable species, e.g. by 
introducing specific impurities that act as inhibitors (Zhang and Nancollas, 

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Crystal growth 283 

Figure 6.44. Six frames from a sequence of time-lapse cinemicrographs showing the 
solvent-mediated transformation of anhydrous Na2SC>4 to Na2SC>4 • H2O at ambient 

Cardew and Davey (1985) proposed a model for solvent-mediated phase 
transformations with which they were able to simulate kinetic data for the 
a— (3 polymorphic transformation of copper phthalocyanine (Honigman and 
Horn, 1973). Solution-mediated transformations have been reported for stearic 
acid (Sato and Boistelle, 1984; Sato, Suzuki and Okada, 1985), magnesium 
phosphate hydrates (Boistelle, Abbona and Madsen, 1983) and L-glutamic acid 
(Kitamura, 1989) and L-histidine (Kitamura, Furukawa and Asaeda, 1994). In 
the cooling crystallization of L-glutamic acid, the metastable a form is the first 
to nucleate and grow, resulting in a crop of 100% a which if separated and 
dried quickly can be maintained indefinitely (Kitamura, 1989). On the other 
hand, if the a form is kept in contact with the crystallization mother liquor the 
solvent-mediated transformation to the (3 form, the stable polymorph, quickly 

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Figure 6.45. Solvent-mediated transformation of vaterite to calcite 

ensues. Davey et al. (1997) have further examined this interesting system and 
described a molecular modelling technique that has led to the identification of 
additives that can selectively inhibit the a — > j3 transformation and allow 
kinetics to dominate the crystallization process so that the stable (3 polymorph 
is suppressed and the metastable a form stabilized. 

Phase transformations can also occur in the solid state, and this mode is 
particularly common in organic solids held close to their melting point. The 
roles of both solid-state and solvent-mediated polymorphic transformations 
have been studied with ammonium nitrate by Davey, Guy and Ruddick (1985) 
and with oleic acid by Suzuki, Ogaki and Sato (1985). 

The industrial process implications of polymorphic crystallizations are dis- 
cussed in section 7.3. 

6.6 Inclusions 

Crystals generally contain foreign impurities, solid, liquid or gas, and the 
pockets of impurity are called 'inclusions'. The term 'occlusion' has also been 
used in this connection, but it has also been applied to the surface fluid that 
becomes trapped between agglomerated crystals and left behind after filtration. 
For this reason, the term 'inclusion' is preferred because it tends to emphasize 
the entrapment of impurity inside a crystal. 

Inclusions are a frequent source of trouble in industrial crystallization. 
Crystals grown from aqueous solution can contain up to 0.5% by mass of 
liquid inclusions, and their presence can significantly affect the purity of 
analytical reagents, pharmaceutical chemicals and foodstuffs such as sugar. 
Inclusions can cause caking (section 7.6) of stored crystals by the seepage of 
liquid if the crystals become broken. 

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Fluid inclusions may often be observed with the aid of a simple magnifying 
glass, although a more detailed picture is revealed under a low-power micro- 
scope with the crystal immersed in an inert liquid of similar refractive index. 
Alternatively the crystal may be immersed in its saturated solution and the 
type of inclusion may be identified by raising the temperature slightly to 
dissolve the crystal: if the inclusion is a liquid, concentration streamlines will 
be seen as the two fluids meet; if it is a vapour, the bubble will rise to the 

A number of terms have been used to describe inclusions, some of which are 
self-explanatory, such as bubbles, fjords (parallel channels), veils (thin sheets of 
small bubbles), clouds (random clusters of small bubbles), negative crystals 
(faceted inclusions) and so on. Most frequently inclusions are distributed 
randomly throughout the crystal, but sometimes they show a remarkable 
regularity, e.g. as in hexamine (Denbigh and White, 1966; Bourne and Davey, 
1977) and ammonium perchlorate (Williams, 1981). Sometimes hour-glass or 
Maltese cross patterns may appear, e.g. as in sucrose (Powers, 1969/70; Man- 
tovani et al., 1985). Several examples of different types of inclusion in crystals 
are illustrated in Figure 6.46. 

Inclusions may be classified as primary (formed during growth) or secondary 
(formed later). Primary fluid inclusions constitute samples of fluid in which the 
crystals grew. Secondary inclusions give evidence of later environments and are 

Figure 6.46. Some examples of liquid inclusions in crystals: (a) a regular pattern in 
ammonium perchlorate, (h) random aligned inclusions in potassium iodide, (c) an L hour- 
glass' pattern in sucrose and (d) 'herring-hone' inclusions in sucrose 

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286 Crystallization 

often formed as a result of crystals cracking due to internal stresses created 
during growth, incorporating mother liquor by capillary attraction and reseal- 
ing later. 

Inclusions readily form in a crystal that has been subjected to dissolution: 
rapid growth occurs on the partially rounded surfaces and entraps mother 
liquor. The rapid healing of dissolution etch pits will do the same. These 
'regeneration' inclusions which usually lie in lines, i.e. along the former crystal 
edges, are characteristic of crystals grown from seeds. 

Interrupted growth generally leads to inclusions. Brooks, Horton and 
Torgesen (1968) attributed the formation of inclusions in ADP and NaC104 
crystals to the introduction of a sudden upward step-change in supersaturation. 
Belyustin and Fridman (1968) suggested that layer growth by the advancement 
of steps across the surface could be used to explain inclusions in KDP and 
NaClO/i. They concluded that the development of an inclusion at any point is 
governed by local conditions, more specifically by the concentration gradient 
along the height of a growth step. A critical step height was postulated beyond 
which a layer of solution can be trapped. 

Large crystals and/or fast growth increase the likelihood of inclusions. 
Denbigh and White (1966) found that crystals of hexamine grew regularly 
when they were quite small, but after they had grown larger than about 
70 urn, cavities began to form at the centre of the faces and these were later 
sealed over to produce a regular pattern of inclusions in the crystals. Cavities 
appeared to form, however, only if the growth rate exceeded a certain value 
when the crystal had reached its critical size. Similar conclusions concerning 
critical size and growth rate criteria have also been recorded for ammonium 
perchlorate (Williams, 1981) and terephthalic acid (Myerson and Saska, 1984). 
The much earlier work of Yamamoto (1939), however, had already identified 
the need to combine both size and growth rate when assessing inclusion 
potential. Some of Yamamoto's recalculated data for NaCl are presented in 
Table 6.2 where it can be seen that the key factor is not the linear, but the 

Table 6.2. Growth rate and appearance of sodium chloride crystals* 

Crystal size L Growth velocity dL/dt Volumetric growth rate' Crystal appearance 
(l(T 4 m) (10- 7 ms-') dV/dt (l(T 15 m 3 s" 1 ) 

1.0 2.9 

1.2 2.9 

1.2 3.8 

2.0 1.4 

1.7 2.6 
1.5 3.7 

1.8 4.1 
2.0 3.7 
3.0 1.8 

* Calculated from data of Yamamoto (1939) 
1 Volumetric rate increase dV/dt = 3L? ■ dL/dt 



















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Crystal growth 287 


1i 2 


o o o 

O J 

o °o 



o / 



o o / 

oo o <y 



o / 



/* •• 


do/ •% • 


/•• •• 

oo om 


- o Am 

• • 





2 4 6 8 

dt/dL ( s m _l x I0 6 ) 

Figure 6.47. Criteria for the avoidance of mother liquor inclusions in sodium chloride 
crystals. (After Yamamoto, 1939) 

volumetric, growth rate that is important. Figure 6.47 suggests another way of 
setting a dividing line between the inclusion and inclusion-free zones. 

The questions of why a cavity often forms at a face centre and why it 
subsequently seals over have been the subjects of much speculation. The work 
of Bunn (1949) and Humphreys-Owen (1949) showed that the supersaturation 
is generally lower at the centre of the crystal face than at the edges, but small 
crystals tend to grow layerwise away from the centres. Bunn's explanation for 
this unexpected finding was that the diffusion field around a small crystal 
would tend to develop spherical symmetry and this results in the component 
of the concentration gradient normal to the face being greater near the centre 
than near the edges, thus causing more solute to be transported to the centre. 
However, when the crystal is large enough, the above situation is reversed 
(Denbigh and White, 1966) and the corners and edges grow more rapidly than 
the face centres, and cavities form. Later, when the face grows beyond a certain 
size, growth layers are generated on the macroface, grow inwards and meet to 
seal the inclusion, as indicated diagrammatically in Figure 6.48 (Murata and 
Honda, 1977; Dzyuba, 1983; Sato, 1988). 

Figure 6.48. Development of a mother liquor inclusion by the interaction of growth layers 
of different height and velocity 

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288 Crystallization 

The adsorption of impurities, including the solvent, on the crystal surface can 
also lead to impeded growth and hence to inclusions. There is evidence that 
crystals reject impurity during growth, and for this reason liquid inclusions may 
be richer in impurities than the mother liquor from which the crystals grew. 
Growth instabilities on crystal faces resulting in dendritic growth can also lead 
to the trapping of mother liquor by the side-branching and impingement of 
dendrites (Myerson and Kirwan, 1977). Solid particles may also be included 
into the crystal particularly at the nucleation stage when foreign bodies act as 
heteronuclei. Saito et al. (1998, 1999) have shown that physical contacts 
between crystals and/or the temporary attachment of small crystal fragments, 
such as attrition nuclei, significantly enhance the growth rate of the particular 
face for a short period during which generated macrosteps interact and entrap 
mother liquor. This work was carried out with sodium chloride crystals, but the 
mechanisms postulated could be of more general application. 

Inclusions may sometimes be prevented if the crystals are grown in the pres- 
ence of certain ionic impurities, e.g. traces of Pb 2+ and Ni 2+ allow near-perfect 
crystals of ADP to be grown for piezoelectric use; traces of Pb 2+ help good 
crystals of NaCl to be grown. Anionic surfactants are particularly effective for 
eliminating inclusions in ammonium perchlorate growth from aqueous solution 
(Hiquily and Laguerie, 1984). A change of solvent may have a significant effect. 
Hexamine, which readily develops inclusions when grown from aqueous solu- 
tion, contains none when grown from methanol or ethanol. An increase in the 
viscosity of the mother liquor may also help; small amounts of carboxymethyl 
cellulose added to the solution have been known to have a beneficial effect. 
Ultrasonic vibrations have also been tried with moderate success. 

Under isothermal conditions inclusions may change shape or coalesce as the 
internal system adjusts itself towards the condition of minimum surface energy. 
If the temperature is raised, negative crystals (faceted inclusion cavities) may be 
formed by a process of recrystallization. Fluid inclusions cannot be removed by 
heating alone. In fact even heating to decrepitation frequently fails to destroy 
all the inclusions. However, liquid inclusions can actually move under the 
influence of a temperature gradient. Since solubility is temperature dependent, 
crystalline material dissolves on the high solubility side of the inclusion, diffuses 
across the liquid and crystallizes out on the low solubility side (Wilcox, 1968). 
Henning and Ulrich (1997) measured migration rates of water inclusions in 
crystal layers of captolactam induced by temperature gradients. Migration 
progressed towards the warm surface at rates proportional to the temperature 
gradient while inclusions increased in size and changed their shape. Large 
inclusions moved faster than small ones. The relevance of these observations 
to industrial solid-layer melt crystallization processes (section 8.4) was dis- 

A general review of inclusions has been written by Deicha (1955). Powers 
(1969/70) and Mantovani et al. (1985) give comprehensive accounts of inclu- 
sions in sugar crystals, and a world-wide coverage of research on inclusions, 
although mainly of geological interest, is provided by the annual COFFI 
(1968 ff) reviews. A practical guide to fluid inclusion studies, with a geological 
bias, has been written by Shepherd, Rankin and Alderton (1985). 

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7 Recrystallization 

7.1 Recrystallization schemes 

It is often possible to remove the impurities from a crystalline mass by dissol- 
ving the crystals in a small amount of fresh hot solvent and cooling the solution 
to produce a fresh crop of purer crystals, provided that the impurities are more 
soluble in the solvent than is the main product. This step may have to be 
repeated many times before a yield of crystals of the desired purity is obtained, 
depending on the nature of the phase equilibria exhibited by the particular 
multicomponent system (Chapter 4). A eutectic system, for example, can yield 
a near-pure crystal product in one step (see Figure 4.4) whereas a solid solution 
system would need many (see Figure 4.7). 

A simple crystallization scheme is shown in Figure 7.1. An impure crystalline 
mass AB {A is the less soluble pure component, B the more soluble impurity) is 
dissolved in the minimum amount of hot solvent S and then cooled. The crop of 
crystals X\ will contain less impurity B than the original mixture: but if the 
desired degree of purity has not been achieved, the procedure can be repeated: 
crystals X\ are dissolved in more fresh solvent S and recrystallized to give 
a crop X<l, and so on. 

In a sequence of operations of the above kind the losses of the desired 
component A can be considerable, and the final amount of 'pure' crystals 
may easily be a minute fraction of the starting mixture AB. This question of 
yield from recrystallization processes is of paramount importance, and many 
schemes have been designed with the object of increasing both yield and 
separation efficiency. The choice of solvent depends on the nature of the 
required substance A and the impurity B. Ideally, B should be very soluble in 
the solvent at the lowest temperature employed, and A should have a high 
temperature coefficient of solubility so that high yields of A can be obtained 
from operation within a small temperature range. Some of the factors affecting 
the choice of a solvent are discussed in section 3.2. 


-X Z 

Figure 7.1. Simple recrystallization 

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A modification of the simple recrystallization scheme in Figure 7.2: additions 
of the original impure mixture AB are made to the system at certain intervals. 
The first two stages are identical with those illustrated in Figure 7.1, but the 
crystals X2 are set aside and fresh feedstock AB is dissolved by heating in 
mother liquor Lj. On cooling, a crop of crystals X3 is obtained and a mother 
liquor L3, which is discarded. Crystals X3 are recrystallized from fresh solvent 
to give a crop X\, which is set aside; the mother liquor L4 can, if required, be 
used as a solvent from which more feedstock can be crystallized. If the crops 
X2, X4, X$, etc. are still not pure enough, they can be bulked together and 
recrystallized from fresh solvent. 

The triangular fractional recrystallization scheme shown in Figure 7.3 makes 
better use of the successive mother liquor fractions than does that of Figure 7.2. 
Again, A and B are taken to be the less and more soluble constituents, 
respectively. The mixture AB is dissolved in the minimum amount of hot 
solvent S, and then cooled. The crop of crystals X\ which is deposited is 
separated from the mother liquor L\ and then dissolved in fresh solvent. The 
cooling and separating operations are repeated, giving a further crop of crystals 
X2 and a mother liquor Lj. The first mother liquor L\ is concentrated to yield 
a crop of crystals X3 and a mother liquor L3. Crystals X3 are dissolved by 
warming in mother liquor L2 and then cooled to yield crystals X5 and liquor 
L5. Crystals X2 are dissolved in fresh hot solvent and cooled to yield crystals 
X4 and liquor L4. Liquor L3 is concentrated to give crystals X& and liquor L(,. 
The scheme can be continued until the required degree of separation is effected. 
The less soluble substance A is concentrated in the fractions on the left-hand 

f l — 

X 3 *" Xe, 

Figure 7.2. Simple recrystallization with further additions of feedstock 


Increasing composition of A Increasing composition of B 

Figure 7.3. Fractional recrystallization of a solution {triangular scheme) 

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Recrystallization 29 1 

\-P 2 

2P[\-P) (\-Pf 3 

3 R? ZP\\-P) 3P(\-P) (I-/T 

Figure 7.4. Analysis of the triangular fractional crystallization 

side of the diagram, the more soluble constituent B on the right-hand side. If 
any other substances with intermediate solubilities were present, they would be 
concentrated in the fractions in the centre of the diagram. 

If the starting material contained a unit quantity of component A, and each 
crystallization step resulted in the deposition of a proportion P of this com- 
ponent, the proportions of A which would appear at any given point in the 
triangular scheme (see Figure 7.4) would be given by a term in the binomial 

[P+(l-P)] n =l (7.1) 

Pn,r= "' v -r-\l-Pf (7.2) 

r\(n — r)\ 

where p nr = proportion of A at a point represented by row r and stage n. Thus, 
for example, 

PX2 = | • ^ (3 - 2) d " Pf 

= 3P(1 - P) 2 

A modification of the triangular scheme is shown in Figure 7.5. In this case 
further quantities of the feedstock AB are added to the system by dissolving it 
in successive mother liquors on the right-hand side of the diagram. This scheme 
is particularly useful if component A has a high temperature coefficient of 

Several other much more complex schemes for fractional recrystallization 
can be used, their aim being to increase the yield of the desired constituent by 
further re-use of the mother liquors. A detailed account of these methods 
has been given by Tipson (1956). Figure 7.6 illustrates two of them. In the 
'diamond' scheme {Figure 7.6a) the outermost fractions are set aside when they 
have reached a predetermined degree of purity, and fractionation is continued 
until all the material is obtained either in a crystalline form or in solution in 
a final mother liquor. If necessary, various crystal fractions can be bulked 

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292 Crystallization 
S -AB 

/ X. 

5 -X, 

Lr AB 

S -X % ^3 


X A L A 

Figure 7.5. Simple recrystallization with further additions of feedstock 

5 V 5 V 

s St ° rtina , c Starting 

•S. material 5 materi J, 

5. Iff Iff S Iff 16 

5 VVl \/\/\ 

\ }\ }K }\ S K 2a 2b 2c 

\ X X X X * 3 ' *' 3/ fc 

4ff 4A 4/- 4,* 4- ), / 1 / \ / \ / > 

S3 5 

X X X X ft 4 * 4 * V 4/ 4* 

1 X X H © — £13 V V V 5/ 

rf V }/ V fe 5 ^e/ V V V V 

uvyw Hsfvvw^ 

* rfVW 5-e/VVVX 

w /\ /\ /\ A 

lOrf 10* icv Off 106 

(a) (b) 

Figure 7.6. Fractional recrystallization schemes: (a) diamond; (b) double withdrawal. 
(After Tipson, 1956) 

together and recrystallized, and in a similar manner the mother liquors can 
receive further treatment. 

Unfortunately, the various fractions obtained by the diamond scheme will 
differ in composition, and relatively pure crystals will be mixed with relatively 
impure ones. This difficulty can be overcome by the use of the 'double- 
withdrawal' scheme shown in Figure 7.6b. The procedure is the same as that 
used in the diamond scheme up to the point where no further fresh solvent is 
used (line 5 in Figure 7.6a). At this point it is assumed that crystals 5a and 
mother liquor 5/ have reached the desired degree of purity, and they are both 
set aside from the system. Fresh solvent is then added to the crystal crop 6b to 
yield a purer crop lb, which is arranged to have a purity similar to that of crop 
5a. Crop lb is set aside, crop 8c crystallized from pure solvent, and so on. 

Theoretical analyses and surveys of the factors affecting the choice of differ- 
ent fractional crystallization schemes have been made by Doerner and Hoskins 

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Recrystallization 293 

(1925), Sunier (1929), Garber and Goodman (1941) and Tipson (1956). Joy and 
Payne (1955) have discussed problems associated with the fractional crystal- 
lization of similar substances where one substance is present in very small 
amount. The use of the Chlopin (1925) and Doerner and Hoskins (1925) 
relationships, which describe the distribution of isomorphous impurities 
between solid and solution phases, is further discussed in section 7.2.3. 

Procedures for the design of industrial fractional crystallization processes for 
multicomponent systems, utilizing appropriate eutectic and solid solution phase 
diagrams, have been described by Fitch (1970, 1976), Chang and Ng (1998), 
Cisternas (1999), Cesar and Ng (1999) and Wibowo and Ng (2000). 

Theoretical stages 

The number of theoretical stages required in a process of fractional crystal- 
lization from solution can be analysed by the well-known McCabe-Thiele and 
Ponchon-Savarit graphical methods commonly used for fractional distillation 
(Matz, 1969). In the Ponchon-Savarit diagram (upper section of Figure 7.7) the 
abscissa records crystal compositions, x, or mother liquor concentrations, y. 
The ordinate represents the solvent-solute mass ratio, N. The system used here 
as an example is lead and barium nitrates in water, which form a continuous 
series of solid solutions with no hydrate formation. The N-x curve, therefore, is 
the abscissa of the diagram (N = 0) and the N—y curve has no discontinuities. 
The region between the N-x and N—y curves represents solid-liquid mixtures. 
As in similar diagrams for liquid-liquid and vapour-liquid systems, this area is 
interlaced with tie-lines whose end-points correspond to the solid (x) and liquid 
(y) phase compositions in quilibrium. 

Equilibrium values from the upper Ponchon-Savarit diagram are used to 
construct the equilibrium curve in the lower McCabe-Thiele diagram as fol- 
lows. For a given tie-line, a vertical line is drawn down from the end-point on 
the N—y curve on to the diagonal (x = y) in the lower diagram. A horizontal 
line is then drawn to the left to meet the vertical from the other end-point of the 
tie-line on the N-x curve. The intersection gives a point on the equilibrium 
curve. This procedure is repeated. 

The inlet and exit streams (F = feed, C = crystals, S = solution) can be 
located on the operating diagram. Points S\ and Co represent the solution 
leaving and the crystal 'reflux' entering the top (stage 1) of the crystallization 
section. Points S n +\ and C„ represent the solution 'reflux' entering and the 
crystals leaving the bottom (stage ri) of the concentration section. The min- 
imum crystal reflux ratio, i?min, is obtained by extending the tie-line through F 
to meet the vertical from Co at P. Then 

distance PS\ 

distance Si Co 

For reflux ratio R > R m i n (where R = QS\ jS\ Co) the operating line passes 
through Fmore steeply than the tie-line and determines the solution and crystal 
'poles' Q and W. Arbitrarily drawn lines radiating from these poles are used in 

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c CT 

Pb-rich crystals 







Ba -rich 

20 40 60 80 

Percent Pb (NOjlj in crystal, x 

7.7. Ponchon-Savarit and McCabe-Thiele diagrams for the fractional crystal- 
of lead and barium nitrates from aqueous solution. {After Matz, 1969) 

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Recrystallization 295 

the construction of the 'crystallization' and 'concentration' curves in the lower 
McCabe-Thiele diagram, in the same way as the tie-lines are used to construct 
the equilibrium curve. 

The continuous fractional crystallization of a mixture of Pb(N03) 2 (the more 
soluble component) and Ba(N03) 2 (the less soluble) is shown diagrammatically 
at the right of Figure 7.7. This scheme is similar to that for fractional distillation 
or countercurrent extraction, but it is not essential to operate the process in 
a column; any arrangement of contact vessels which provides the necessary 
stage-wise countercurrent contact of solid and liquid phases would suffice. In 
the scheme depicted Ba(N03) 2 crystallizes out in the upper 'crystallization' 
section, above the feed entry point, and Pb(N03) 2 dissolves in the lower 
'concentration' section. The crystals, which progress downwards, are therefore 
enriched in Ba, and the solution, which progresses upwards, is enriched in Pb. 
The Ba-rich crystals are removed through a filter and the filtrate is returned to 
the column as 'reflux'. The Pb-rich solution leaving the top of the column is 
evaporated to yield Pb-rich crystals, some of which are returned as 'reflux' at 
the top of the column. 

To produce crystals containing 95 per cent Ba(N03) 2 at the top and 95 per 
cent Pb(NC>3) 2 at the bottom from a 50 per cent feedstock, using a reflux ratio 
of 1.36, four theoretical stages would be adequate as shown. The upper 
Ponchon-Savarit diagram could also be used to determine the number of 
theoretical stages. 

7.2 Resolution of racemates 

The resolution of racemic mixtures, i.e., the separation of at least one of the 
enantiomers in pure form, is an important step in the manufacture of chiral 
products in the pharmaceutical, agrochemical, flavour and perfumery indus- 
tries and for other specialty chemicals. The necessity to resolve racemic mix- 
tures is particularly strong in the pharmaceutical industry since the two 
component enantiomers (see section 1.9 for definition of terms) in the equi- 
molar mixture can have very different pharmacological activities. Thalidomide 
is a frequently quoted example: one stereoisomer was the beneficial agent for 
preventing morning sickness in pregnant women while the other caused serious 
birth defects. 

A crystalline racemate, may be either a conglomerate (an equimolal physical 
mixture of two enantiomorphs) or a racemic compound (two enantiomers 
homogeneously distributed in the crystal lattice). Conglomerates, the much 
rarer (< 10%) of the two types of racemate are easier to resolve. The mixture 
of d and l crystals can sometimes be distinguished under a microscope and in 
very rare cases can even be separated by hand, as in the case of sodium 
ammonium tartrate (section 1.9.1). In practice, however, manual sorting is 
impossible and resolution by crystallization is the usual means of producing 
at least one of the d or l enantiomers in pure form. The d, l system of 
molecular classification, which is still commonly encountered, is being used 
here as an alternative to the more rigorous r, s system (section 1.9) since the 

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symbols R and S are used in the following phase diagrams to denote racemate 
and solvent respectively. 

The phase equilibria for a racemic mixture can be represented on a binary 
phase diagram with composition as the abscissa and temperature as the ordin- 
ate. Figure 7.8a represents a conglomerate system and Figure 7.8b indicates the 
formation of a racemic compound. Both of these diagrams are symmetrical, 
unlike those for non-chiral systems (as in Figures 4.4 and 4.5). For example, 
a conglomerate system has one eutectic point, at the 50 : 50 d : l composition. 
Any solution to the left of it can produce pure d crystals and solutions to the 
right yield pure l crystals. On the other hand, the racemic compound system 
has two eutectic points equidistant from the 50% mark representing the com- 
position of the racemic compound which cannot be resolved into either of the 
pure enantiomers in a single crystallization operation. 








Figure 7.8. Phase diagrams: (a) and (c) for conglomerates; (b) and (d) for racemic 
compounds, (c) and (d) illustrate the potential existence of very narrow zones of terminal 
solid solutions {After Jacques, Collet and Wilen, 1981) 

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Recrystallization 297 

At this point an explanation of the term 'pure' might be helpful. For many 
industrially bulk-produced organic chemicals a purity of >95% is often 
accepted as justifying the designation 'pure'. For some specialty chemicals a 
purity of > 99% may be demanded. For purities > 99.9% the term 'ultra pure' 
is frequently applied. For many chiral products, an enantiomeric purity of 
around 98% can be acceptable since up to about 2% of the other enantiomer 
usually has little or no adverse effect on the activity of the product. In any case, 
a crystallization operation cannot produce 100% pure crystals for a variety of 
reasons, e.g., they can be contaminated with residual solution- or reaction-based 
impurities that have not been removed by washing, or have been incorporated 
into the crystal interstitially or as liquid inclusions, and so on. Furthermore, 
account should be taken of what has been called terminal solid solutions, which 
inevitably accompany both eutectic and chemical compound systems, as 
shown in Figures 7.8b and 7.8c where the zones of miscibility (solid solutions), 
are identified by the shaded areas which are greatly enlarged for ease of 
visualization (Jacques, Collet and Wilen, 1981). These very narrow zones are 
generally impossible to detect with any certainty since the melting point or 
solubility data needed to draw the phase diagram are insensitive to such small 
quantities of impurity. 

A mixture of d and l enantiomers in the presence of a solvent S constitutes a 
ternary system in which the equilibria are best represented on a triangular 
diagram (see section 4.6.3). The effects of temperature on solubility and phase 
change can also be included. Figure 7.9a shows the transition of a racemic 
compound R, stable at the lower temperatures t\ and ti, to a conglomerate at 
the higher temperature tj. 

Phase diagrams such as those depicted in Figures 7.8 and 7.9 can readily be 
constructed by measuring the solubilities of a range of mixtures of the two 
enantiomers and determining their equilibria by methods described in section 
4.5. At all times during equilibrium determination undissolved solid must 



Figure 7.9. Resolution of racemates: (a) isotherms on a ternary diagram (temperature 
h < h < ?3); (b) simple recrystallization procedure for a conglomerate. (S = solvent, R = 
racemic composition, 50% d : l) 

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remain in agitated contact with solution. Equilibration in such systems can 
take some considerable time and it is recommended that samples are analysed 
at 1-day intervals until equilibrium is achieved, preferably from both the over- 
and under-saturated conditions (section 3.9). Subsequent analyses of the 
equilibrium saturated solutions and their corresponding solid phases can be 
made by the appropriate techniques described in section 3.9.1. 

A simple recrystallization procedure for the resolution of conglomerate 
systems is shown in Figure 7.9b. A convenient starting point is point M, 
representing a metastable aqueous solution containing the racemate R and 
a slight excess of enantiomer L. This solution may first be prepared at an 
elevated temperature and then cooled to the phase diagram temperature. At 
point M the solution is supersaturated with respect to both enantiomers, but 
more supersaturated with respect to L. Enantiomer L is induced to crystallize 
alone by adding a few L seeds and the mother liquor composition moves from 
M to N following the line LMN to the extent that at N its optical rotation is 
roughly of equal magnitude, but opposite sign, to that at M. The now dextro- 
rotatory mother liquor is then separated from the L crystals and used to 
dissolve an equivalent mass of racemate R to yield, by heating and subsequent 
cooling, a metastable supersaturated solution O. D seeds are then added to 
induce crystallization of enantiomer D and the mother liquor composition 
changes from O towards P. At P enantiomer D is recovered, more racemate 
R is dissolved in the mother liquor, and the cycle is repeated. A possible flow 
scheme, with a recycle loop, for the continuous seeded crystallization of a 
conglomerate system is shown in Figure 7.10. 

Examples of different crystallization procedures have been given by Secor 
(1963); Collet, Brienne and Jacques (1980); Jacques, Collet and When (1981); 
Asai (1983); Samant and Chandalla (1985); Sheldon (1993); Collins, Sheldrake 
and Crosby (1995). 

Racemic compound systems, which account for more than 90% of all 
racemic mixtures, cannot be resolved by direct crystallization, but a common 


50:50 D:L , 

D-Rich mother liquor 



with D seeds 

Crystallize with L seeds 

D Crystals 


L Crystals 

Figure 7.10. Flow diagram for the continuous crystallization of a conglomerate system. 
(After Stahly and Byrn, 1999) 

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Recrystallization 299 

route to resolution is to react the 50 : 50 mixture of the (+) and (— ) enantiomers 
with an optically pure resolving agent, e.g., an acid or base (most usually an 
amine), to give a mixture of diastereomers which are non-enantiomeric (++) 
and ( — h) and hence have different physical properties. For example, a race- 
mate of an acidic substance A with, say, the dextro form of an optically active 
base B will give 

(±)-A + (+)-B -> (+)-A ■ (+)-B + (-)-A • (+)-B 

and the two salts (+)-A • (+)-B and (— )-A • (+)-B can be separated by 
fractional crystallization. The phase diagram of such a mixture is generally 
asymmetric and crystallization will yield crystals enriched in one of the isomers. 
Practical examples have been given by Jacques, Collet and Wilen (1981); 
Yokota et al. (1998) and Stahly and Byrn (1999). 

Crystallization from the melt (usually at low temperature) is also a possibility 
for recovering enantiomers. A temperature-composition phase diagram for 
a conglomerate is the same as that for a eutectic system containing two com- 
ponents D and L (section 4.3.1). The special feature of the conglomerate 
diagram is its symmetry, which arises from the thermodynamic identity of the 
D and L enantiomers (melting point, enthalpy of fusion, etc.) giving the eutectic 
an equimolal racemic composition R (Figure 7.11). The following procedures 
are possible. 

The liquid conglomerate is supercooled, following a path from say X through 
E to point Y below the solidus. The two liquidus curves for D and L may be 
extended (dotted lines) beyond E into the metastable region. At point Y, seeds 
of one of the enantiomorphs are added. L seeds, for example, cause L crystals 
to be deposited and the residual liquid becomes richer in D. By the mixture rule 
(section 4.3), one mole of racemate gives MY/MO mole of pure L and YO/MO 

C R L 


Figure 7.11. Recrystallization of a conglomerate from the melt 

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300 Crystallization 

mole of residual liquid of composition c. The L crystals are then separated and 
the D-rich liquid is seeded with D to give a crop of D crystals. This process of 
alternate L and D seeding can be repeated. 

The yield from each crystallization depends on the ratio MN/PN, which 
increases with the supercooling EY. It is, of course, necessary to find the 
optimal conditions of temperature and agitation which allow maximum yields 
of D and L without disturbing the metastable conditions on which the process 
depends (Collet, Brienne and Jacques, 1980). 

Sublimation (section 8.3) could have a role to play in the separation of 
enantiomers since the vapour phase arising from a solid mixture of enantiomers 
is always racemic, irrespective of the ratio of enantiomers present in the solid. 
Therefore, if the starting solid contains unequal amounts of enantiomer, sub- 
limation will increase the purity of the major component in the residual solid 
phase. There does seem to be a possibility for improving the purity of near-pure 
enantiomers, e.g. those that had been partially resolved by other methods. 
However, as only a few experimental studies of enantiomer separation by 
sublimation have so far been recorded, the technique must still be regarded as 
speculative (Jacques, Collet and Wilen, 1981). 

7.3 Isolation of polymorphs 

The ability of a single compound to crystallize in more than one crystallo- 
graphic form (polymorphism) is encountered in a wide range of industries 
including pharmaceuticals, dyestuffs, agrochemicals, photochemicals, and 
other specialty compounds, both organic and inorganic. Similarly, it may also 
be possible to crystallize several different solvates, e.g., hydrates, which 
although not strictly speaking polymorphs can also be included in the general 
considerations outlined below. 

As described in section 6.5, the different polymorphs of a given substance, 
although chemically identical, exhibit different physical properties that can 
considerably affect the end-use of the material. For example, one polymorph 
of a pharmaceutical compound may have quite significant differences in bio- 
availability and pharmacological action from another. It is essential, therefore, 
when investigating a new chemical substance with a view to eventual industrial 
production, to investigate at an early stage the possibility of the existence of 
different polymorphs or solvates and how they may be isolated. Without such 
an investigation, new phases may make a sudden appearance years after the 
development of the first form and this can often be very inconvenient. How- 
ever, as Bavin (1989) has pointed out, the existence of different polymorphs or 
solvates need not always be considered as a problem. Indeed, if discovered early 
enough, they may have potential commercial advantages in allowing increased 
patent coverage and greater flexibility in, for example, the formulation of 

For monotropic polymorphs {Figure 6.42a) only one form is fhermodyn- 
amically stable at all temperatures in the range considered. All other forms are 
metastable and potentially capable of transforming into the stable form. For 

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Recrystallization 301 

enantiotropic polymorphs (Figure 6.42b) two or more forms are thermodyn- 
amically stable over different temperature ranges. In these cases it is essential to 
identify the transition temperatures as accurately as possible since they mark 
boundaries that should be avoided when planning processing operations. 
Furthermore, compounds with transition temperatures in the ambient range 
are likely to cause problems in subsequent storage and use. 

A number of guidelines for the laboratory procedures that can be adopted in 
the preliminary search for possible polymorphs or solvates have been proposed 
Bavin (1989) and Nass (1991). These include: crystallizing from a wide range of 
solvents (polar, non-polar, hydrophilic and hydrophobic) at different temper- 
atures; chilling saturated solutions rapidly; precipitation by rapid quenching 
with a liquid non-solvent; heating excess solid with a high boiling solvent; 
crystallization from the melt or by sublimation, and so on. The identity and 
purity of all product crystals should then be checked by appropriate analytical 
techniques (McCauley, 1991). See also section 3.9.1. 

Once the polymorphs or solvates have been isolated and identified, samples 
can be used as seed crystals in subsequent crystallization operations to promote 
the production of a specific form. Seeding is particularly necessary in cases 
involving the so-called concomitant polymorphs (Bernstein, Davey and Henk, 
1999), the phenomenon of different polymorphs crystallizing simultaneously 
(section 6.5). It is important to identify any process conditions that could 
result in polymorphic transformation, e.g., the initiation of a solvent-mediated 
transformation in monotropic systems during a drying operation. Solid 
phase transformations can be temperature dependent, but they can also occur 
during energetic processes such as grinding or tabletting (Pirttimaki et ah, 

The process implications of polymorphism in organic compounds has been 
considered by Nass (1991) who outlined some general recommendations for the 
batch cooling crystallization of a desired polymorph, either stable or meta- 
stable. First, it is necessary to isolate and identify each polymorph and to 
generate solubility data in more than one solvent in order to determine if the 
system is monotropic or enantiotropic. This information will help in the selec- 
tion of a solvent for the industrial process, although the ultimate choice will 
also have to include considerations of process yield, solvent recovery costs, 
hazards, etc. 

For enantiotropic systems the temperature range of the crystallization pro- 
cess will be determined by the particular polymorph required. If it is metastable 
below the transition temperature, crystallization should begin just above trans- 
ition temperature, where the kinetics are relatively slow. Seeding with the 
desired polymorph at this point is recommended. If the required polymorph 
is stable below the transition temperature, seeding should be commenced just 
below the transition temperature. 

Seeding is also recommended for monotropic systems. For the stable poly- 
morph, the temperature after seeding should be held constant for a predeter- 
mined time to allow the solution to desupersaturate, after which an appropriate 
cooling profile (section 7.5.5) should be adopted to maintain a constant 
controlled supersaturation. If the metastable polymorph is required, and the 

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solution is supersaturated with respect to the stable form, the solution should 
be seeded with the metastable form and cooled rapidly to avoid any solvent- 
mediated transition. As noted above, the drying conditions for metastable 
polymorphs must be chosen carefully to avoid any solvent-mediated trans- 
formation occurring. 

7.4 Recrystallization from supercritical fluids 

At temperatures and pressures above the critical point, where liquid and 
vapour phases are no longer distinguishable (section 4.2) supercritical fluids 
(SCFs) exhibit very different properties from those of their corresponding 
liquid and gaseous phases. For example, the dielectric constant of liquid water 
(~81) falls to around 2 for supercritical water at which value, since it no longer 
acts as a polar solvent, it can dissolve many organic compounds and provide 
a medium for recrystallization. On the other hand inorganic salts are virtually 
insoluble in supercritical water. Solute solubilities in supercritical fluids can 
undergo considerable changes with relatively small changes in pressure, and at 
constant pressure they generally pass through a minimum with temperature. 
These characteristics give the possibility of separating different solutes, by 
manipulating pressure or temperature in the so-called 'cross-over' region 
{Figure 7.12). Rapid depressurization, for example, creates high supersatura- 
tion, fast nucleation rates and consequently large numbers of small crystals. 
Chang and Randolph (1989) showed how small (< 1 urn) uniform particles of 
/3-carotene, a vitamin A precursor, could be precipitated by rapidly expanding 
a solution in supercritical ethylene (critical point, 10 °C and 50 bar) through 
a nozzle. 


Figure 7.12. Solubility curve minima exhibited by two different solutes A and B in 
a supercritical fluid 

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Recrystallization 303 

Of all the solvents considered in recent years as possible SCFs for crystal- 
lization processes, the only two that now command any notable attention are 
water and CO2 primarily because they are non-flammable, non-toxic, low-cost 
and readily available. Of these, CO2 is attracting greater support on account of 
its more accessible critical point (31 °C and 74 bar) compared with that of water 

One problem with SCFs as crystallization solvents is that the solubilities of 
most organic compounds are generally low. Whilst the supersaturations devel- 
oped can appear to be high when expressed as the ratio c/c* (section 3.12.1), 
they are usually low when expressed as a mass concentration driving force 
(c — c*) and the consequent low yields and productivity can make the process 
appear commercially unattractive. Nevertheless, SCF crystallization could still 
be profitable in, for example, the separation of isomers and polymorphs and 
the purification of high-value products such as pharmaceuticals. Attempts have 
been made to increase solubility in SCFs by adding small amounts of a low 
molecular weight organic liquid, e.g. an alcohol or ketone, as a co-solvent 
(Dobbs, Wong and Johnston, 1986) although this can present subsequent 
downstream separation problems in an industrial process. 

Some recent examples of exploratory crystallizations using supercritical CO2 
have been described by Mohamed, Debendetti and Prud'homme (1989); Kelley 
and Chimowitz (1989); Sako, Satu and Yamane (1990); Berends, Bruinsma 
and van Rosmalen (1993); Liu and Nagahama (1996); Tai and Cheng (1997); 
Shekunov, Hanna and York (1999). 

Attention has recently been drawn to the possible advantages of 'nearcritical' 
rather than supercritical water as a benign solvent for both organic and ionic 
compounds (Eckert, Liotta and Brown, 2000). Nearcritical water is defined as 
liquid water at temperatures between about 250 and 300 °C and at its corres- 
ponding vapour pressure. Like ambient water, nearcritical water can hydrate 
ions, but it can also dissolve non-polar organic compounds. Polar organics are 
virtually insoluble in nearcritical water making it a good solvent in which to 
conduct chemical syntheses. Whilst no crystallization applications have yet 
been reported, the potential does appear to exist. 

7.5 Zone refining 

For the removal of the last traces of impurity from a substance, fractional 
crystallization from the melt cannot be applied with any degree of success. 
Apart from the fact that an almost infinite number of recrystallization steps 
would be necessary, there would only be a minute quantity of the pure sub- 
stance left at the end of the process. High degrees of purification combined with 
a high yield of purified material can, however, be obtained by the technique 
known as zone melting or zone refining, originally developed by Pfann (1966) 
for the purification of germanium for use in transistors. Purification by zone 
refining can be effected when a concentration difference exists between the 
liquid and solid phases that are in contact during the melting or solidification 
of a solid solution. The method is best explained by means of a phase diagram. 

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Figure 7.13a shows a phase diagram for two substances A and B that form a 
simple solid solution: in this case the melting point of A is higher than that of B. 
When a homogeneous melt M of composition x is cooled solid material will 
first be deposited at some temperature T. The composition of this solid, which 
is in equilibrium with the melt of composition x (point L on the liquidus), is 
given by point S* on the solidus. As cooling proceeds, more solid is deposited, 
and the concentration of B in the solid increases towards S. A similar reasoning 
may be applied to the reversed melting procedure starting from temperature T', 
where a liquid of composition L* is in equilibrium with a solid S of composition 
x. The solidification of a homogeneous molten mixture M yields a solid with an 
overall composition x, but owing to segregation the solid mixture is not 
homogeneous. Adjustments in the composition of the successive depositions 
of solid matter will not take place because of the very slow rate of diffusion in 
the solid state. The further apart the liquidus and solidus lines are, the greater 
will be the difference in concentration between the deposited solid and residual 

A measure of the expected efficiency of separation is given by a factor known 
as the segregation coefficient k. The significance of this coefficient can be seen 
in Figure 7.13b, where the liquidus and solidus for a binary solid solution in the 
regions of low B and low A concentrations are represented by straight lines. As 
zone melting is only useful for the refining of substances with low impurity 
contents, these are the regions that are of interest. For dilute solutions the 
segregation coefficient k is defined by 

concentration of impurity in solid 

concentration of impurity in liquid 
slope of liquidus / 
slope of solidus s 

(at equilibrium) 


In Figure 7.13b it can be seen that k\ = l\js\ and kj = I2/S2, and also that 
s\ > l\ and Si < h. In general, therefore, it may be said that 










s 2 \^ 




Figure 7.13. Phase diagram for simple solid solution: (a) solidification of a mixture; 
(b) liquidus and solidus drawn as straight lines in regions of near-pure A and B 

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Recrystallization 305 


Direction of solidification 
Figure 7.14. Molten bar of unit length undergoing progressive directional solidification 

k < I when the impurity lowers the melting point 


k > I when the impurity raises the melting point 
When k < I, the impurity will concentrate in the melt, when k > 1 in the 
solidifying mass. The nearer the value of k approaches unity, i.e. the closer 
the liquidus and solidus approach, the more difficult the segregation becomes. 
When k = 1, no zone refining is possible. 

The impurity concentration at any point along a solid bar or ingot that was 
originally molten and then progressively cooled and solidified along its length 
(Figure 7.14) can be expressed in terms of the segregation coefficient. The 
concentration C of impurity (solute) in the solid at the solid-liquid interface 
is given by 

C = kC, (7.5) 

where Q is the concentration of impurity in the liquid phase at the interface. 
At any distance g along the bar of unit length, the concentration C at the 
solid-liquid interface is given by the equation 

C = fcC (l - g) k ~ X (7.6) 

where Co is the initial concentration of the impurity in the homogeneous 
molten bar. 

The distribution of impurity along a bar subjected to this process of direc- 
tional solidification can be calculated from equation 7.6. The greater the devi- 
ation of k from unity, the greater the concentration gradient along the bar. 

7.5.1 Constitutional supercooling 

When a crystal grows in a melt or solution, impurity is rejected at the solid- 
liquid interface. In a stagnant fluid, e.g. with the crystalline phase gradually 
advancing into an unstirred melt, the impurity may not be able to diffuse away 
fast enough, in which case it will accumulate near the crystal face and lower the 
melting point in that region. Thus the effective temperature driving force, 
AT = T* — T, is decreased and the crystal growth rate is retarded. However, 
the melting point in the main bulk of the melt remains unaffected, so AT" 
increases away from the interface. 

This condition, known as constitutional supercooling, represents an instabil- 
ity and the advancing face usually breaks up into finger-like cells, which 
progress in a more or less regular bunched array. In this manner heat of 
crystallization is more readily dissipated and the tips of the projections advance 

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306 Crystallization 

clear of the concentrated impurity and grow under conditions of near-maximum 
driving force. Impure liquid may be entrapped in the regions between the fingers, 
and a succession of regular inclusions may be left behind. At low constitu- 
tional supercooling, i.e. in a relatively pure system, the advancing interface is 
generally cellular. At high constitutional supercooling (high impurity) dendritic 
branching may occur. This sort of behaviour is common in, for example, the 
casting of metals. 

7.5.2 Zone purification 

Directional solidification is not readily adaptable for purification purposes, 
because the impurity content varies considerably along the bar at the end of the 
operation. The end portion of the bar, where the impurity concentration is 
highest, could be rejected and the process repeated after a remelting operation, 
but this would be extremely wasteful. 

The technique known as zone melting does lend itself to repetitive purifica- 
tion without undue wastage. In this process a short molten zone is passed along 
the solid bar of material to be purified. If k < 1, the impurities pass into the melt 
and concentrate at the end of the bar. If k > 1, the impurities concentrate in the 
solid; thus in this case a cooled solid zone could be passed through the melt. 
The sequence of operations for molten zone refining is shown in Figure 7.15a,b. 
Although the impurity concentration in the bar, after the rejection of the high- 
impurity end portion, is much lower than initially, there is a considerable 
concentration gradient along the bar. In order to make the impurity concentra- 
tion uniform along the bar, the process known as zone levelling must be 
employed; further zoning is carried out in alternate directions until a homo- 
geneous mid-section of the bar is obtained {Figure 7.15c). Both the high- and 
low-impurity ends of the bar are discarded. 

7.5.3 Zone refining methods 

A few of the basic arrangements used in zone refining (Parr, 1960) are shown in 
Figure 7.16. Figure 7.16a shows the material contained in a horizontal crucible 
along which a heater is passed; or the crucible may be pulled through a 
stationary heater, and the molten zone travels through the solid. Several zones 
may be passed simultaneously at fixed intervals along the bar {Figure 7.16b) in 
order to reduce purification time. It is essential that the heaters be spaced at a 
sufficient distance to prevent any spread of the molten zones; for materials of 
high thermal conductivity and substances that exhibit high degrees of super- 
cooling, alternate cooling arrangements may have to be fitted between the 
heaters. The ring method {Figure 7.16c) permits a simple multi-pass arrange- 
ment. The vertical tube method {Figure 7.16a) is useful when impurities that 
can sink or float in the melt are present. A downward zone pass can be used in 
the former case, an upward pass in the latter. 

Different methods of heating may be used to produce molten zones; the 
choice of a particular method is usually governed by the physical characteristics 

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Recrystallization 307 



'A I st pass 

»W^ ^»^ ^^^i 2nd pass 






3rd pass 


Initial molten zone Passage of zone Concentrated impurity zone 



after several 




Homogeneous vg; 

Figure 7.15. (a) and (b): A 3-pass zone refining operation; (c) zone levelling 


••• M* ••• 




Figure 7.16. Simple methods of zone refining: (a) horizontal, single-pass; (h) horizontal 
concurrent passes; (c) continuous concurrent passes in 'broken ring'' crucible; (d) vertical 
tube, upward or downward passes. (After Parr, 1960) 

of the material undergoing purification. Resistance heating is widely employed, 
and direct-flame and focused-radiation heating are quite common. Heat can be 
generated inside an ingot by induction heating, and this method is often used 
for metals and semiconductors. Heating by electron bombardment or by elec- 
trical discharge also have their specific uses. Solar radiation, focused by lenses 
or reflectors, affords an automatic method for zone movement, on account of 
the sun's motion. 

The speed of zoning is a factor that can have a considerable effect on the 
efficiency of zone refining. The correct speed is that which gives a uniform zone 
passage and at the same time permits impurities to diffuse away from the 


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308 Crystallization 

solidifying face into the melt. If the zone speed is too fast, irregular crystal- 
lization will occur at the solidifying face, and impure melt will be trapped 
before it can diffuse into the bulk of the moving zone {Figure 7.17a). For the 
purification stages, the speed of zoning can vary from about 50 to 200mmh~ 1 
and speeds of about 5 to 15mmh _1 are common for zone levelling. 

The tube or crucible that contains the material undergoing purification 
should not provide a source of contamination and must be capable of with- 
standing thermal and mechanical stresses. The purified solid must be easily 
removable from its container, so the melt should not wet the container walls. 
The choice of the container material depends on the substance to be purified. 
Glass and silica are commonly used for organic substances, and silica for many 
sulphides, selenides, arsenides and antimonides; graphite-lined silica is often 
used for metals. 

External contamination can be prevented by dispensing with the use of 
a conventional container. For example, a vertical zone refining technique that 
uses a 'floating zone' is shown in Figure 7.17b. The bar of material is fixed 
inside a container tube without touching the walls and the annulus between the 
bar and tube may contain a controlled atmosphere. Surface tension plays 
a large part in preventing the collapse of the molten zone, but the control of 
such a zone demands a high degree of experimental skill. 

Although the largest applications of zone refining are in the fields of semi- 
conductors and metallurgy, the technique has been used very successfully for 
the purification of a large number of chemical compounds. Herington (1963), 
for example, reports the particular application of zone refining to the prepara- 
tion of very pure organic compounds. Nicolau (1971) has described the puri- 
fication of inorganic salts by zone refining in the presence of a solvent, which 
allows operation at temperatures much lower than the melting point. He calls 
the technique zone-dissolution-crystallization, and describes the production of 
reagent-grade di-sodium phosphate, ammonium alum and copper sulphate 
from impure commercial-grade chemicals. 

Comprehensive accounts of zone refining and its applications have been 
given by Pfann (1966), Schildknecht (1966), Zief and Wilcox (1967), Shah 
(1980) and Sloan and McGhie (1998). 




y 10/\ 


Too fast 







Figure 7.17. (a) Influence of zone speed; (b) floating-zone technique 

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Recrystallization 309 

7.6 Single crystals 

There is a large demand nowadays for single, pure and defect-free crystals of 
innumerable substances, e.g. for the scientific appraisal of the chemical and 
physical properties of pure solids, for use in solid-state electronics, where 
crystals of certain substances are required for their dielectric, piezoelectric, 
paramagnetic and semiconductor properties, and for optical and laser materials 
and synthetic gem-stones (see Table 7.1). 

The production of large single crystals demands exacting techniques, but 
broadly speaking there are three general methods available, viz. growth from 
solutions, from melts, and from vapours. There are many variations, however, 
in the processing techniques available within these three groups. 

This subject is somewhat outside the main theme of the present book, so only 
a brief summary will be attempted here. However, several comprehensive 
accounts have been made of this specialized aspect of crystallization practice 
(Laudise, 1970; Faktor and Garrett, 1974; Pamplin, 1980; Hurle, 1993). 

7.6.1 Low temperature solution growth 

Slow crystallization from solution in water or organic solvents has long been 
a standard method for growing large pure crystals of inorganic and organic 
substances. Basically, a small crystal seed is immersed in a supersaturated 
solution of the given substance and its growth is regulated by a careful control 
of the temperature, concentration and degree of agitation of the system. For 
instance, a tiny selected crystal may be mounted on a suitable support and 
suspended in a vessel containing a solution of the substance, maintained at 
a fixed temperature. Slow rotation of the vessel will give an adequate movement 
of the solution around the crystal, and slow, controlled evaporation of the 
solvent will produce the degree of supersaturation necessary for crystal growth. 
Alternatively, the vessel may remain stationary and the growing crystal, or 
several suitably mounted crystals, may be gently rotated in the solution. 

The actual operating conditions vary according to the nature of the crystalliz- 
ing substance and solvent; the optimum supersaturation and solution movement 

Table 7.1. Some uses of single crystals 

Uses Substances 

Piezoelectric quartz, Rochelle salt, ammonium 

and potassium dihydrogen phosphates, 
ethylenediamine tartrate, triglycine phosphate 

Optical materials CaF 2 , a-SiC>2 

Lasers ruby, sapphire 

Paramagnetic materials a-Al 2 03, TiC>2, CaF 2 

Semiconductors Ge, Si, GaAs 

Luminescence and ZnS and CsS activated with Ag, 

fluorescence Tl, Cr, Mn and Cu salts; various organic crystals 

Gem-stones alumina (sapphire and ruby), diamond 

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310 Crystallization 

past the crystal must be found by trial and error. Generally speaking, the degree 
of supersaturation must not be high, and in any case the solution must never be 
allowed to approach the labile condition. The degree of supersaturation should 
be kept as constant as possible to ensure a constant rate of deposition of solute 
on the crystal seed. Holden (1949) states that, for salts with solubilities in the 
approximate range 20-50 per cent by mass, the maximum linear growth rate 
that can be tolerated by the fastest-growing faces of a crystal is about 1-3 mm 
per day; faster rates tend to give imperfections. Single crystals are better grown 
by a cooling than by an evaporation process, as supersaturation can then be 
much more closely controlled. 

A typical example of a cooling-type growth unit is shown in Figure 7.18a. 
The carefully selected seed crystals are supported on a rotating arm, which 
is submerged in the gently agitated system, and grow under carefully control- 
led conditions. A very slow cooling rate is used, often less than 0.1°Ch~'. 
Torgesen, Horton and Saylor (1963) have given a detailed description of an 
apparatus and procedure for growing large single crystals by this method. 
Single crystals can also be grown by holding them in a fixed position in a 
growth cell of the type described in section 6.2.2 (Figure 6.12) through which 
solution of the required level of supersaturation flows at a controlled steady 

7.6.2 High temperature solution growth 

Many substances normally considered insoluble in water have an appreciable 
solubility at elevated temperatures and pressures. This property is utilized in the 
technique called 'hydrothermal crystallization', which is basically crystalliza- 
tion from aqueous solution at high temperature (350-550 °C) and pressure (1-3 
kbar). The operation is carried out in a steel autoclave (Figure 7.18b), which 
can be provided with a silver or platinum liner for protection. The technique 
has proved satisfactory for the growth of silica and aluminosilicates from 
aqueous alkaline solutions. Quartz crystal, the ideal piezoelectric material, is 
now grown in this manner on a commercial scale. 

Molten salt solvents may also be used for high-temperature solution growth 
in the technique normally called 'flux growth'. Oxide crystals, like yttrium 
aluminium garnet, Y3AI5O12, for example, may be grown from ionic solvents 
such as PbO— PbF2— B2O3, while metallic solvents are suitable for covalently 
bonded semiconductors such as Si, GaAs and InP, while diamond can be 
crystallized from molten Ni-Fe alloys. 

Hydrothermal and flux growth and their industrial applications have been 
the subjects of several comprehensive reviews (Lobachev, 1971; Elwell and 
Scheel, 1975; Elwell, 1980; Wanklyn, 1983). 

7.6.3 Growth in gels 

A room-temperature method, particularly useful for the production of single 
crystals of substances that are thermally unstable or have a very low solubility 
in water, is the technique of growth in a gel. The gel provides a kind of 

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Recrystallization 3 1 1 


Stainless steel 

Seed crystals 

/^.Crushed quartz 




Figure 7.18. Crystallization from solution: (a) temperature-lowering methods, (b) hydro- 
thermal growth 

protective barrier for the growing crystals; it eliminates turbulence, yet it 
imposes no strain on the crystals, and permits a steady diffusion of the growth 
components. The diffusion and reaction processes are retarded and very much 
larger crystals are grown than could be produced by normal reaction and 
precipitation techniques. 

One technique for growth by controlled diffusion in a gel may be illustrated 
by the growth of calcium tartrate tetrahydrate crystals by reacting calcium 
chloride with D-tartaric acid in a silica gel (Rubin, 1969). A freshly made 
solution of sodium metasilicate (21. 6g in 250cm 3 of water, density 1.034, 
acidified to pH 3.5 with normal tartaric acid solution) is poured into test-tubes 
(6x1 in, two-thirds full) sealed to prevent loss of water and allowed to set over 
a period of 3 days at 40 °C. A normal solution of calcium chloride is then 
carefully pipetted on to the gel and nucleation immediately takes place at the 
gel-nutrient interface. After several hours single crystals appear below the 
interface and grow until they can no longer be supported by the gel. Crystals 
from 2 to 10 mm have been grown by this method over a period of a few weeks. 
Doped crystals can also be grown by doping the nutrient solution. 

Surveys of growth in gels have been made by Henisch (1988) and Arora (1981). 

7.6.4 Growth from the melt 

A single crystal can be grown in a pure melt in a manner similar to that 
described earlier for growth in a solution. For example, a small crystal seed 
could be rotated in a supercooled melt, or the melt could flow past a fixed seed 

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in a growth cell. However, these methods do not find any significant applica- 
tion. The most widely used techniques for the production of single crystals from 
the melt can be grouped into four categories: the withdrawal or pulling tech- 
niques, the crucible methods, flame fusion and zone refining. The latter method 
has been described in some detail above. 

The pulling techniques are typified by the Kyropoulos and Czochralski 
methods. In both cases a small, carefully selected seed crystal is partially 
immersed in a melt, kept just above its melting point, and the growing crystal 
is maintained just below its melting point by holding it on a water-cooled rod or 
tube (see Figure 7.19). The apparatus is usually kept under reduced pressure or 
supplied with an inert atmosphere. Silicon, germanium, bismuth, tin, alumi- 
nium, cadmium, zinc, intermetallic compounds, and many organic substances 
have been grown as large strainfree single crystals in this manner. 

The main difference between the methods of Kyropoulos (see Figure 7.19a) 
and Czochralski (see Figure 7.19b) is that in the former the seed is permitted to 
grow into the melt, while in the latter the seed is withdrawn at a rate that keeps 
the solid-liquid interface more or less in a constant position. Pull rates depend 
on the temperature gradient at the crystal-melt interface and can vary from 1 to 
40mmh _1 . The steeper the gradient the faster the growth rate and, hence, the 
faster the permissible rate of withdrawal. 

Compounds that dissociate on melting cannot be grown by the vertical pull- 
ing technique unless means are provided for suppressing dissociation. A simple 
elegant means is provided by the liquid encapsulation technique (Mullin, 


Cooling water 

Inert gas 


Figure 7.19. Crystal pulling and withdrawal techniques: (a) Kyropoulos, (h) Czochralski 

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Recrystallization 313 

Straughan and Bricknell, 1965) in which the melt surface is covered by a floating 
layer of transparent liquid. The encapsulant, e.g. boric oxide, B2O3, a low- 
melting-point glass, acts as a liquid seal provided that the inert gas pressure on 
its surface is greater than the dissociation pressure of the compound. As 
the crystal is pulled from the melt, a thin film of encapsulant adheres to its 
surface and suppresses dissociation. The technique is particularly useful for the 
growth of semiconductor and metal crystals; even highly dissociable compounds 
such as GaP (vapour pressure 35 bar at 1470 °C) can be grown successfully on 
a commercial scale by this technique. 

An example of the crucible method is that due to Bridgman and Stockbarger 
(see Figure 7.20a). The melt is contained in a crucible with a conical bottom 
which is lowered slowly from a hot to a cold zone. The two sections of the 
furnace, which are kept at about 50 °C above and below the melting point of 
the substance, respectively, are isolated by a thermal shield. The bulk of the 
solidified material, apart from the tip of the cone where the nuclei gather, is 
a single crystal. This method has proved successful for many semiconductor 
materials and alkali halides. 

The flame fusion technique (see Figure 7.20b) was originally devised in 1904 
by Verneuil for the manufacture of artificial gemstones, such as corundum 
(white sapphire) and ruby. This method is now used for the mass production 
of jewels for watches and scientific instruments. A trickle of fine alumina 
powder plus traces of colouring oxides is fed at a controlled rate into an 
oxyhydrogen flame. Fusion occurs and the molten droplets fall on to a ceramic 
collecting rod. A seed crystal cemented to the rod is fused in the flame and the 
rod is lowered at a rate that allows the top of the growing crystal (known as 
a boule) to remain just molten. Renewed interest has recently been shown in this 
method for the production of rubies for lasers. 

Melt growth techniques have been reviewed by Pamplin (1980), Sloan and 
McGhie (1998) and Hurle (1993). 

7.6.5 Growth from vapour 

Crystal growth from a vapour by direct condensation, without the intervention 
of the liquid phase, can be used to produce small strain-free single crystals of 
substances that sublime readily. Large single crystals cannot be grown by this 
method. A number of techniques are available. For example, a gas stream, such 
as N2 or H2S, may be passed over a sublimable substance, such as cadmium 
sulphide, in a heated container. The vapours then pass to another part of the 
apparatus, where they condense, in crystalline form, on a cold surface. 

Sublimation in sealed tubes is also used for the preparation of single crystals 
of metals, such as zinc and cadmium, and non-metallic sulphides. A quantity of 
the material is placed at one end of the tube, along which a temperature 
gradient is maintained, so that sublimation occurs at the hot end and crystal- 
lization at the other. An electric furnace with a number of independently 
controlled windings is used to maintain the temperature gradient to give a rate 
of sublimation sufficiently slow for the grown crystals to be single and not 

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314 Crystallization 

Kanthol winding- 
Inconel lining 

Mullite tubing 


Kanthal leads 

Platinum crucible 
and lid 

Polished platinum 
radiation baffle 

■ Crystal 

-Alundum layer 

Metal base 


@===a Tapper 

Oxygen — = 

Oxide p 

\ ^ 
\( Screen 

Oxide powder 



c ^|- Firebrick 
— Window 

Ceramic rod 


,1^ Lowering 
tt© mechanism 

Figure 7.20. (a) The Bridgman-Stockbarger technique; (h) the Vemeuil houle furnace 

Vapour phase growth techniques have been reviewed by Faktor and Garrett 
(1974), Pamplin (1980), Hurle (1993) and Krabbes (1995). 

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8 Industrial techniques and equipment 

8.1 Precipitation 

Precipitation is a widely used industrial process. It is also a very popular 
laboratory technique, especially in analytical chemistry, and the literature on 
this aspect of the subject is voluminous (see Kolthoff et cil, 1969). Precipitation 
plays an important role not only in chemistry but also in metallurgy, 
geology, physiology, and other sciences. In the industrial field, the manufacture 
of photographic chemicals, pharmaceuticals, paints and pigments, polymers 
and plastics utilizes the principles of precipitation. For the production of 
ultrafine powders precipitation may be considered a useful alternative to 
conventional crystallization followed by milling. 

The term precipitation very often refers to nothing more than fast crystal- 
lization, although sometimes it also implies an irreversible process, e.g. many 
precipitates are virtually insoluble substances produced by a chemical reaction, 
whereas the products of most conventional crystallization processes can usually 
be redissolved if the original conditions of temperature and solution concentra- 
tion are restored. Another distinguishing feature of precipitation processes is 
that they are generally initiated at high supersaturation, resulting in fast 
nucleation and the consequent creation of large numbers of very small primary 
crystals. Although precipitation, like all crystallization processes, consists of 
three basic steps (supersaturation, nucleation and growth) two subsequent 
secondary steps usually have a profound effect on the final crystalline product. 
The first is agglomeration, which generally occurs soon after nucleation, and the 
second is ageing, a term used to cover all irreversible changes that take place in 
a precipitate after its formation. 

A common method for producing a precipitate is to mix two reacting 
solutions together as quickly as possible, but the analysis of this apparently 
simple operation is exceedingly complex. Primary nucleation does not neces- 
sarily commence as soon as the reactants are mixed, unless the level of 
supersaturation is very high, and the mixing stage may be followed by an 
appreciable time lag before the first crystals can be detected. This event, 
which is often referred to as the end of the induction period, depends on the 
supersaturation, temperature, efficiency of mixing, intensity of agitation, 
presence of impurities, and so on. As explained in section 5.5, however, an 
experimentally measured induction period is not identical with the critical 
nucleation time since critical nuclei have to grow before they can be detected. 
Primary nucleation, both homogeneous and heterogeneous, together with 
growth of the nuclei, may continue for some time until sufficient crystal 
surface area has been created to cause a rapid desupersaturation. This event, 
which will be referred to as the end of the latent period, can be virtually 
identical with the experimental induction period at high supersaturations 

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316 Crystallization 

(see Figure 5.12). Secondary nucleation is probably a rare event in precipitat- 
ing systems. 

Some indication of the particular nucleation and/or growth mechanisms 
predominating during the early stages of precipitation may be gained by plot- 
ting measured induction periods as a function of (1) (logS)~ , (2) (logS) - 
and (3) log(5 _1 ). A straight line obtained from any of these plots can give 
an indication of the predominant mechanism. For example, (1) suggests 
either homogeneous nucleation or primary nucleation followed by diffusion- 
controlled growth, (2) indicates polynuclear growth, and (3) screw-dislocation 
growth. Further support for such mechanisms, however, must be sought by 
evaluating a relevant physical property from the slope of such functions 
(e.g. interfacial tension, see section 5.5) and assessing its validity (Sohnel and 
Mullin, 1978b, 1988a; Sohnel and Garside, 1992). 

8.1.1 Agglomeration 

Small particles in liquid suspension have a tendency to cluster together. Such 
terms as 'agglomeration', 'aggregation', 'coagulation' and 'flocculation' have 
all been applied in this area, although without any generally accepted rules of 
definition. For simplicity, therefore, the term 'agglomeration' will be used 
exclusively in this section. 

Interparticle collision may result in permanent attachment if the particles are 
small enough for the van der Waals forces to exceed the gravitational forces, 
a condition that generally obtains for sizes <lum. Smoluchowski (1918) 
showed that the half-time, (*, the time needed to have the number of particles 
in a monodisperse system, may be expressed as 

t* = [n t /(n -n t )]t (8.1) 

where n t and «o are the number of particles present at time t and in the original 
monodispersion (t = 0), respectively. For a binary collision process in a water- 
dispersed phase, in which all collisions are effective, Walton (1967) suggested 
that not* ~2 x 10 11 . If an agglomerated system were arbitrarily defined as one 
in which more than 10% of the particles have agglomerated in less than 1000 s, 
then aqueous systems containing less than 10 7 particles/cm 3 can be said to be 
non-agglomerating. The rather interesting conclusion to be drawn from this is 
that agglomeration may not be expected in systems where heteronucleation 
occurs since the number of hetero nuclei is generally much less than 10 7 cm~ 3 , 
in fact it can be less than 10 4 cm~ 3 . On the other hand, agglomeration is 
quite common in systems that have nucleated homogeneously when n often 
exceeds 10 7 cm~ 3 . Of course, not all interparticle collisions result in permanent 
contact and in lyophobic systems charge stabilization greatly decreases the rate 
of agglomeration. 

An over-simplified but graphic example of the interactive effects of super- 
saturation, nucleation and growth in the development of precipitated par- 
ticles is given by Fiiredi-Milhofer and Walton (1969), who considered the 
homogeneous nucleation of three different systems at an arbitrary value of 

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Industrial techniques and equipment 3 1 7 

supersaturation S (= c/c*, see section 3.12.1) = 100. The three solubilities, 
c* are 10~ 7 ,10~ 4 and lO^'molL -1 , respectively, so the respective solution 
concentrations, c, at the onset of nucleation are 10 5 ,10 2 and lOmolL -1 . If 
the number of particles nucleated is say 10 6 cm~ 3 , it may be concluded that the 
maximum particle sizes expected from such precipitations are 1,10 and 100 (im, 
respectively. Of course, this is a very crude calculation, but the results do show 
an order of magnitude agreement with practical experience. 

Another way to look at the problem is to estimate the approximate nucleated 
particle size if the solutions of the above three substances were all of the same 
concentration, e.g. 1 molL -1 . The supersaturation of the first solution is S = 10 7 
and this would nucleate homogeneously, forming particles around 1 nm and 
producing a colloidal system (gel). These small primary particles would ulti- 
mately agglomerate, but the gel could remain stable for long periods if left 
undisturbed. The second solution (S = 10 4 ) would also nucleate homogeneously 
forming primary particles around 0. 1 (im which again would agglomerate easily, 
but a conventional precipitate would rapidly develop. The third solution 
(S = 10) would probably nucleate heterogeneously, yielding crystals around 
10 (xm which could, under favourable conditions, remain discretely dispersed. 

Smoluchowski (1918) distinguished between two types of agglomeration for 
colloidal particles in suspension: 

1. Perikinetic (static fluid, particles in Brownian motion) 

2. Orthokinetic (agitated dispersions). 

Both modes can occur in precipitation processes, but in a stirred precipitator 
orthokinetic agglomeration clearly predominates. From the Smoluchowski 
kinetic expressions for perikinetic and orthokinetic agglomeration it can be 
deduced (Sohnel and Mullin, 1991) that the relationship between agglomerate 
size D and time / may be expressed by 

D 3 (t) = A x + Bit (perikinetic) (8.2) 


\ogD(t) = A 2 + B 2 t (orthokinetic) (8.3) 

where A and B are particle-fluid system constants. Figure 8.1 shows plots of 
these two equations for the early stages (first few minutes) of barium tungstate 
precipitation in an agitated vessel. The linear plot for equation 8.3, and not for 
equation 8.2, demonstrates that the agglomeration was orthokinetic rather than 

Equations 8.2 and 8.3 are strictly applicable only to the early stages of 
agglomeration since they imply an unlimited increase of agglomerate size with 
time, which is unrealistic. Indeed, an upper limiting agglomerate size D max is 
often reached, after a certain time and in many cases this can be comparable 
with the Kolmogoroff microscale of turbulence (typically around 25 (im in 
stirred vessels). Z) max is also a function of the mixing intensity and often satisfies 
a relationship (Tomi and Bagster, 1978) of the form 

Z) max oc R-" (8.4) 

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318 Crystallization 



0.25 - 


Figure 8.1. Plots of equations 8.2 [O] and 8.3 [9] for data from the precipitation of barium 
tungstate at an initial concentration of 3 mol m~ 3 . D is the median particle size (\im). (After 
Sdhnel and Mullin, 1991) 

where R is the stirrer speed and exponent n acquires a value of or 3 
when viscous or inertial forces, respectively, acting on the agglomerated 
particles are dominant. In the transition region n adopts some intermediate 

An example of the time-development of a precipitate is shown in Figure 8.2 
which traces the particle size distribution of strontium molybdate precipitated 
batchwise by mixing aqueous solutions of SrCi2 and Na2Mo04. In this 
particular case the final steady-state distribution produced under agitated 
conditions is approached in around five minutes after mixing the reactants 
and remains constant up to 1 hour. The final size distribution and the 
behaviour of the primary particles are also influenced by the agitator speed 
in the reaction vessel and the reactant solution concentrations, but an indica- 
tion of the development can be seen in the selected photomicrographs (taken 
from different runs) in Figure 8.3. Towards the end of the induction period 
(~15s) the crystals appear mainly as discrete spheres (Figure 8.3a). Within 
about 1 min., however, the crystals become elongated (Figure 8.3b). After 
about 30 min., virtually all the crystals are agglomerated (Figure 8.3c). 

Useful surveys of agglomeration kinetics have been made by Nyvlt et al. 
(1985), Sohnel and Garside (1992) and Wachi and Jones (1995). Recent papers 
on the subject by Collier and Hounslow (1999) and Zauner and Jones (2000a) 
are also noteworthy. 

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Industrial techniques and equipment 319 



_l I I I hi IX 

uL i i n iin 

60 min after mixing 

I I I I I I ' ' I 

,1 i ujir I l2^ 

6 min ifter mixing 
I I I ll I I I I 1,1 II I 

10 1 after mixing 

-J- ' ' ' I "I I ' I I I I I 

1 10 100 1000 

Particle size, urn 

Figure 8.2. Volume percentage particle size distribution o/SrMo0 4 particles precipitated 
batchwise as a function of the time after mixing the reacting solutions. {After Sohnel, 
Mullin and Jones, 1988) 

Figure 8.3. Particles of precipitated SrMo0 4 observed by SEM (a) towards the end of the 
induction period, (b) after 1 min. and (c) after 30 min. {After Sohnel, Mullin and Jones, 

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320 Crystallization 

8.1.2 Ageing 


When solid particles are dispersed in their own saturated solution there is a 
tendency for the smaller particles to dissolve and the solute to be deposited 
subsequently on the larger particles. Thus the small particles disappear, the 
large grow larger and, theoretically, the particle size distribution should ulti- 
mately change towards that of a monosized dispersion. The reason for this 
behaviour lies in the tendency of the solid phase in the systems to adjust itself to 
achieve a minimum total surface free energy. This process of particle coarsening 
was first called 'Ostwald ripening' by Liesegang (1911), after the proposer of 
the mechanism (Ostwald, 1896). 

The driving force for ripening is the difference in solubility between small and 
large particles, as given by the size-solubility (Gibbs-Thomson) relationship 
(equation 3.58) which for the present purpose may be written as 




2lV (8.5) 


where v = molar volume of the solute, c(r) = solubility of small particles of size 
r, c* = equilibrium saturation for large particles (r — > oo), 7 = interfacial ten- 
sion and v = number of ions in a formula unit. As discussed in section 3.7 
a significant increase in solubility occurs when r < 1 um. From equation 8.5, 
expanding the logarithm for c(r)jc* ~ 1 (ripening takes place at very low super- 
saturations), we get 

c(r) - c* « -I— (8.6) 


If mass transport is possible between the particles in a polydisperse precipitate, 
the large will grow at the expense of the small. If the growth kinetics are first- 
order, diffusion-controlled, then for a change in particle radius with time 
(see equation 6.37) 

dr Dv[c - c(r)] 
at r 

where c is the average bulk solution concentration, and from equation 8.6 

dr _Dv 

At r 

(c — c ) 


where c — c* is always positive during precipitation. Setting equation 8.8 equal 
to zero, it follows that all particles of size 

vVLT(c — c*) 

are in equilibrium with the bulk solution (dr/dt = 0). All particles smaller than 
this will dissolve dr/dt < 0), and all particles larger will grow (dr/dt > 0). 

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Industrial techniques and equipment 321 

The speed with which ripening occurs depends to a large extent on the 
particle size and solubility. For example, small crystals of a moderately soluble 
salt such as K2SO4 with a solubility c\ ~ 10 3 molm~ 3 will age much faster than 
those of the almost insoluble BaSC>4 (c* 2 ~ 10~ 2 molm -3 ) if kept in solutions of 
the same supersaturation, say S = c/c* = 1.1. The driving force, for mass 
transfer Aci = (S — Y)c\ = 10 2 and Aci = 10~ 3 molm~ 3 , and if a growth law 
Rq = KqAc" is assumed (equation 6.14), with the growth coefficients not 
differing by more than an order of magnitude, the relative growth rates of the 
two salts are ~10 5 : 1 for first-order growth kinetics rising to ^lO 12 : 1 for 

For diffusion-controlled growth kinetics it can be shown (Nielsen, 
1964; Hanitzsch and Kahlweit, 1969) that the linear growth velocity approx- 
imates to 

dr 7V 2 Z)c* 

dt 3isRTr 2 
which on integration gives 


vRTr 3 
7V z Z)c* 

thus the smaller the particle size, r, or the higher the solubility, c* , the faster 
ripening process. 

Of course, the ripening process, which occurs at very low supersaturation, is 
probably more likely to be controlled by a surface reaction than by a diffusion 
process, and under these circumstances ripening could be considerably 
retarded. For surface reaction growth kinetics the linear growth velocity obeys 
a relationship such as 

f ( = k(c-c*T (8.12) 

where A; is a growth rate constant, and if an arbitrary value of n = 2 is assumed, 

\7vc*/ k 
Therefore, from equations 8.11 (t = ?d) and 8.13 (/ = /r) 

Substituting typical values in equation 8.14 for a relatively insoluble salt such 
as PbS0 4 (T= 300 K, D = lO^n^s" 1 , 7= 10~' Jrrr 2 , R = 8.3 Jmok 1 K" 1 , 
c* = 10~' molm -3 , k = 10~ 6 m 7 mol~ 2 s~', v = 2), we get Zr ~ tv, which 
indicates that the interfacial reaction can exert a dominant effect on the ripen- 
ing process. Any additive to the system which slows down the surface reaction 
step will automatically retard the ripening process and thus stabilize the 

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322 Crystallization 

suspension. Substances such as gelatin and carboxymethyl cellulose are com- 
monly used for this purpose. 

Ripening can change the particle size distribution of a precipitate over a 
period of time, even in an isothermal system, but the change can be accelerated 
by the use of controlled temperature fluctuations. This process, known as 
'temperature cycling', has been utilized to alter the physical characteristics of 
organic and inorganic precipitates (Carless and Foster, 1966; Nyvlt, 1973; 
Brown, Marquering and Myerson, 1990). 

Kahlweit (1975) has made a comprehensive analysis of ripening mechanisms. 
Sugimoto (1978) has critically assessed relationships between diffusion- 
controlled and reaction-controlled ripening and reviewed the experimental 
determination of the respective kinetic constants. 

Phase transformation 

Although Ostwald ripening is rightly regarded as an important ageing process 
for a precipitate that remains in contact with its mother liquor, particularly if 
the primary crystals are smaller than 1 um, it is not the only ageing process that 
can be encountered. Nor is it always the most important one. 

Another frequent occurrence is the first-precipitation of a metastable phase 
(Ostwald's rule of stages - section 5.7) followed by a phase transformation to the 
final product. The metastable phase may be an amorphous precipitate, a poly- 
morph of the final material, a hydrated species or some system-contaminated 

Magnesium hydroxide, precipitated by mixing aqueous solutions of MgCl2 
and NaOH, affords a particularly good example of a complex ageing process 
(Mullin et al., 1989a). After the first few seconds the mean agglomerate size 
(~20 um) decreased steadily with time. Mechanical breakdown of the agglom- 
erates, under agitated conditions, at first suspected, was not responsible 
because the precipitate specific surface area also decreased with time. The 
paradox was shown to be the result of the first-precipitation of a complex 
species (probably MgOH • OC1 • 2H2O) which subsequently dehydrated and 
decomposed to yield Mg(OH) 2 . The consequent agglomerate shrinkage caused 
the primary platelet crystals (^40 nm) to cement together resulting in a specific 
area reduction. 

Calcium sulphate in water exhibits a phase transition near 100 °C between 
the dihydrate (gypsum) and the hemihydrate initiated by the dissolution of 
gypsum and the subsequent nucleation and growth of hemihydrate (Nancollas 
and Reddy, 1974). Dissolution-recrystallization mechanisms have been advanced 
for many similar transformations involving hydrates, e.g. calcium phosphates 
(Nancollas and Tomazic, 1974; Brecevic and Fiiredi-Milhofer, 1976) and for 
polymorphic transformations, e.g. copper phthalocyanine (Honigmann and 
Horn, 1973). Solid-state transformations, frequently encountered in poly- 
morphic changes, are occasionally encountered in hydrate transitions, e.g. 
calcium oxalate (Gardner, 1976). 

Polymorphs and their transformations are discussed in section 6.5. 

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Industrial techniques and equipment 


Precipitate morphology 

The morphological development of a precipitate is a complex combination of 
a variety of processes, including nucleation, growth, habit modification, phase 
transformation, ripening, agglomeration, and so on. The dominant system 
parameters are supersaturation and the level of active impurities, although in 
some aqueous systems pH can also exert a profound effect. 

The dominant influence of supersaturation on the particle size characteristics 
of a precipitate has been summed up in the so-called Weimarn laws of preci- 
pitation (Weimarn, 1926) which, whilst open to criticism from a theoretical point 
of view, still give useful guidelines for batch precipitation behaviour. The 'laws' 
cannot be expressed concisely, but Figure 8.4 assists in their interpretation: 

1. As the concentration of reacting substances in solution is increased, i.e. as 
the initial supersaturation is increased, the mean size of the precipitate 
particles (measured at a given time after mixing the reactants) increases to 
a maximum and then decreases. As the time at which measurement is made 
is increased, the maximum is displaced towards lower initial supersatura- 
tions and higher mean sizes. 

2. For a completed precipitation, the precipitate mean size decreases as the 
initial supersaturation is increased. 

In addition to confirming the well-known beneficial effect of using reason- 
ably dilute reactants to produce coarse precipitates, the laws demonstrate that 
excessive dilution can be detrimental, a fact that is not always fully appreciated. 
Experimental evidence for the Weimarn laws has been provided by Mullin and 
Ang (1977) for the precipitation of nickel ammonium sulphate. 

Some measure of control over nucleation and growth, and hence of precip- 
itation, may also be exercised by the addition of substances, such as surfactants 


Figure 8.4. Weimarn 's 'laws' of precipitation. Time t\ < t2 < tj, 

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and polyelectrolytes. Impurities in the system, whether deliberately added or 
already present, can have a powerful influence on the morphology of the final 
precipitated particles. 

Walton (1967) and Fiiredi-Milhofer (1969) has described how, due to the 
build-up of impurities in the crystal face boundary layer, the predominant 
growth of crystal corners operating in a periodic mode can lead to the develop- 
ment of dendritic-like structures (Figure 8.5). Particles of this type, commonly 
seen for example in precipitates such as Bayer hydrate (alumina hydrate), give 
the impression of being developed by the partial agglomeration of small plate- 
lets (an explanation frequently found in the literature), but the true mechanism 
is probably one of periodic growth (Gnyra, Jooste and Brown, 1974). Another 
type of precipitated particle that is often confused with a true agglomerate is the 
one that results from crystalline out-growths emanating from different surfaces 
of an original heteronucleus. Subsequent growth leads to the development of a 
particle that looks deceptively like an agglomerate. 

Another interesting growth form exhibited by several substances is the 
development of regular isometrical shapes towards the end of the precipitation 
process. The phenomenon was described for the case of barium sulphate by 
Melikhov and Kelebeev (1979) who proposed the mechanism that, in a strongly 
supersaturated solution, homogeneous nucleation followed by growth yields 
isotropic crystals of about 2 nm in size. Their surfaces then develop zones with 
enhanced growth rates which lead to the formation of anisotropic needles, 


Figure 8.5. Representation of the development of a crystal dendrite: (a) the corners, which 
are not blocked by impurity, grow preferentially; (b) periodic growth leads to the 
development of a dendrite-type crystal. (After Walton, 1967) 

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Industrial techniques and equipment 325 

the rapidly growing faces of which have different crystallographic orientations. 
The needle crystals then adhere, first by end to end contacts, forming ring 
structures which subsequently, i.e. at a lower supersaturation, agglomerate in 
a side to side fashion to form isometrical cellular particles. 

It often appears that, for a given system, there is a certain level of super- 
saturation for maximum agglomeration. It is as if under these conditions the 
crystal surfaces become 'sticky'. It is interesting to speculate on the link 
between this behaviour and the character of the adsorption layer (loosely 
bonded, partially integrated groups of the crystallizing species) surrounding 
a growing crystal. At very low supersaturations, the adsorption layer will be 
thin and strongly attracted to the crystal face and the growth units will be 
readily incorporated at the growth sites. At very high supersaturations the 
adsorption layer will be thick and the outer layers of growth units will only 
be loosely attracted to the parent crystal. Under these conditions fluid shearing, 
as the crystal moves through the solution, can sweep portions of the loosely 
bound growth units into the solution where, because they are already partially 
integrated, they become nuclei. However, at some intermediate level of super- 
saturation the adsorption layer may be neither 'thick' nor 'thin' and contact 
between two crystals in the same condition could rapidly result in a permanent 
bond through their integrated adsorption layers. 

Amorphous precipitates 

A substance capable of exhibiting crystallinity is sometimes precipitated in an 
amorphous form. Kolthoff et al. (1969) have suggested that there is a competi- 
tion between the 'aggregation' and 'orientation' velocities of the molecular 
species concerned and the former sometimes dominates, although this is 
undoubtedly an oversimplification. 

Aggregation (non-crystalline) and orientation (crystalline) are both influ- 
enced by supersaturation and temperature, but the type of species is also very 
important. For example, strongly polar salts, such as lead iodide, silver chloride 
and barium sulphate, invariably precipitate in crystalline form. Carbonates of 
calcium and barium and hydroxides of magnesium and zinc do likewise, but 
there is evidence in some of these cases of the prior precipitation of hydrated 
short-lived precursor phases (Mullin et al., 1989; Brecevic and Nielsen, 1990). 
Hydrous oxides like hydrous ferric oxide (o-ferric hydroxide) are generally 
amorphous, especially when precipitated from cold solution. 

Amorphous precipitates can transform, often quite rapidly, into a crystalline 
product on ageing, but the ageing process can sometimes be accompanied by 
chemical change. Hydrous ferric oxide affords such an example: a polymer 
containing eight atoms of Fe agglomerates into a chain structure containing 
more than 50 atoms of Fe which dehydrates on ageing to a yellow amorphous 
ferric acid with a ring structure and finally yields crystalline goethite. 

In recent years there has been an increasing realization that manufactured 
crystalline products can contain small amounts of molecular disorder which 
can be highly detrimental to product quality, particularly in the pharma- 
ceutical industry. Amorphous regions in a crystal are thermodynamically 

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326 Crystallization 

unstable and, if located near the surfaces of water-soluble substances, the 
particles can readily absorb atmospheric water vapour and initiate undesir- 
able physical and chemical changes. Buckton and Darcy (1999) have 
critically reviewed some of the analytical techniques, including vapour sorp- 
tion, DSC, powder X-ray diffraction and IR spectrometry, that can be used 
to characterize partially amorphous materials. An interesting method has 
been proposed by Kishishita, Kishimoto and Nagashima (1996) for interpret- 
ing X-ray diffraction patterns to provide information on a property described 
as 'crystal texture'. The two quantities evaluated are the degree of crystal- 
linity and the preferred orientation of molecular units, both of which are 
shown to be influenced by the rate of crystallization or grinding the dry 
crystalline product. 

8.1.3 Co-precipitation 

To some extent, all precipitates are contaminated with materials originating 
from solution in the mother liquor. The general term 'co-precipitation' may be 
used to cover the many different types of impurity incorporation that can 
occur, including the surface adsorption and lattice entrapment of foreign ions 
and solvent molecules, the physical inclusion of pockets of mother liquor, and 
so on. 

Adsorption phenomena 

The adsorption of salts having an ion in common with the precipitate roughly 
follows the Paneth-Fajans-Hahn adsorption rule which postulates that the less 
soluble the salt, the more easily is it incorporated into a precipitate. For 
example barium chloride is more readily adsorbed by barium sulphate than is 
barium iodide, which is much more soluble than the chloride. 

The dissociability of the adsorbed salt is also important. Adsorption 
decreases as the degree of dissociation of the adsorbed salt increases. The 
deformability of the foreign ion also has an influence. With anions, which are 
generally more easily deformed than cations, the deformability increases with 
size. Dyestuff anions, for example, are highly deformable and this property 
is utilized in the application of adsorption indicators used in volumetric 

Salts having no ion in common with the precipitate can also be incorporated. 
The ions in a lattice surface exhibit residual valence forces and will readily 
attract foreign ions of opposite charge. Some complex 'surface exchange' reac- 
tions may occur (Kolthoff et ai, 1969). For example, when a barium sulphate 
precipitate is shaken with a dilute solution of lead perchlorate, the lead ions are 
strongly attracted by the barium sulphate since lead sulphate is slightly soluble 
and it fits into the barium sulphate lattice. On the other hand, the perchlorate 
ions show no pronounced tendency to adsorb since barium perchlorate is quite 
soluble. Thus, both Ba 2+ , Pb 2+ and SO 2 . - are strongly attracted by the BaSCM 
lattice surface whereas CIO4 is not, and the following surface exchange reaction 
takes place: 

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Industrial techniques and equipment 327 
BaS0 4 + Pb 2+ + 2C10 4 ^ PbS0 4 + Ba 2+ + 2CIO4 

(surface) (solution) (surface) (solution) 

As a result, Ba 2+ in the surface is partly replaced by Pb + and an equivalent 
amount of Ba 2+ enters the solution. This exchange adsorption occurs even 
though the solubility product of lead sulphate is not exceeded. 

Precipitates originating from colloidal solutions (sols or gels) are also liable 
to be contaminated with the counterions. For example, a colloidal solution of 
silver iodide may be prepared by mixing dilute solutions of silver nitrate and 
potassium iodide. The colloidal particles are stabilized by adsorbed I and an 
equivalent amount of K + (the counterion) is present. The Agl sol may be 
flocculated by the addition of an electrolyte, e.g. KNO3, but the precipitate 
will be contaminated with adsorbed KI which is difficult to remove by washing. 
However, counterions can often be readily replaced by other ions. For example, 
if the above contaminated Agl precipitate is washed with dilute H2SO4, K + 
may be replaced by H + . Many such examples are available in textbooks of 
analytical chemistry (e.g. see Kolthoff et al., 1969). 

Solid solution formation 

Mixtures of components that exhibit solid solution behaviour cannot be sep- 
arated in a single step as can, for example, simple eutectic systems. Multistage 
or fractional precipitation schemes must therefore be employed (section 7.1). 
The distribution of an impurity between the solid (i.e. solid solution) and liquid 
phases may be represented by the Chlopin (1925) equation: 


where a and b are the amounts of components A and B in the original solid, 
x and y are the amounts of A and B in the crystallized solid and a—x and b—y 
are the amounts of A and B retained in the solution. D is a distribution 
coefficient. Alternatively, the logarithmic Doerner-Hoskins (1925) equation 
may be used: 

ln(a/x) = Xln (b/y) (8.16) 

The constant A has been called a heterogeneous distribution coefficient to 
distinguish it from the homogeneous distribution coefficient D in equation 
8.15. Under ideal conditions D = X. 

If component A is the impurity, A > 1 indicates that the impurity will be 
enriched in the precipitate. Conversely, if A < 1 it will be depleted. A schematic 
diagram of the effect of precipitation rate on A is shown in Figure 8.6 (Walton, 
1967). In enrichment systems, A — > A e = Z) e as the precipitation rate tends to 
zero. For fast rates of precipitation A — > 1. For depletion systems, an analogous 
situation exists with A — > A^ = D& for very slow precipitation and A — > 1 for 
rapid precipitation. 

Both the Chlopin and Doerner-Hoskins relationships have been widely used 
to correlate the results of fractional precipitation and recrystallization schemes 



- X 

= D 




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328 Crystallization 

Infinitely slow 

Very rapid 

Infinitely slow 

20 40 60 80 

% Major component precipitated 


Figure 8.6. The effect of precipitation rate on the distribution coefficient X (equation 8.16). 
(After Walton, 1967) 

(see section 7.1), although neither is entirely satisfactory from a theoretical 
point of view. 

General problems associated with the fractional precipitation of mixtures of 
similar substances when one is present in a very small amount have been 
discussed by Joy and Payne (1955) and Gordon, Salutsky and Willard (1959). 

Incorporation of solvent 

Solvent molecules are frequently found in association with precipitated mater- 
ials. For example, crystalline substances often form with water molecules 
located at specific sites, e.g. water of crystallization, held in co-ordination 
complexes around lattice cations. Extraneous inclusion of water molecules 
can occur if a co-precipitated cation carries solvation molecules with it. Massive 
incorporation of solvent, together with other soluble impurities can occur in 
random pockets (inclusions) as a result of the physical entrapment of mother 
liquor. Fast crystal growth, leading to growth instabilities, dendrite formation, 
crystal agglomeration, etc., can all contribute to this undesirable feature. An 
account of liquid inclusions in crystals is given in section 6.6. 

8.1.4 Precipitation diagrams 

The inherent complexities of precipitation in multicomponent systems influence 
the system equilibria, nucleation and growth mechanisms and precipitate char- 
acteristics. The graphical procedures developed by Tezak (1966) and Fiiredi 
(1967) afford an interesting approach towards an understanding of the under- 
lying principles. If two precipitating components, e.g. with interacting anions or 
cations, are mixed in increasingly dilute solution, a limit will eventually be 

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Industrial techniques and equipment 


•5 - 

_ -4 


*- 3 


1 1 1 

"T / 

s ^Solubility 
\jr boundary 




. . . \ Q~ \ / 

/ - 

- Precipitation >w *' /\ 
boundary X^^ \ 


/ \ 

/ \ 


/ \ 


/ \ 

" A/ D P 

/ A 

/ A 



-1 -2 -3 -4 -5 -6 

Log (OxZVmol L"' ) 

Figure 8.7. Precipitation boundary (24 h) for calcium oxalate at 25 °C. P = plates, 
D = dendrites, A = agglomerates 

reached where no precipitate will appear, even after a very long time. Tezak 
called this limit the 'precipitation boundary'. A range of reactant concentra- 
tions is normally covered. For example, the concentration of one component 
can be varied while the concentrations of all the others are kept constant. In 
this way, the maximum concentration attainable before the onset of precipita- 
tion, i.e. the precipitation boundary, may be determined. 

The precipitation boundary for some arbitrary 'long period of time', e.g. 
24 hours, may be drawn on the diagram as shown in Figure 8.7. The solubility 
curve is also depicted and its contours reflect any complexing that occurs 
in solution and the resulting equilibria. The zone between the two curves 
represents the condition of metastability. The dotted diagonal representing 
equivalent concentrations of the initially added precipitating components, 
divides the precipitation diagram into two symmetrical parts, in each of which 
one of the precipitating components is in excess. Any line parallel to the 
equivalence line represents a different constant ratio of the precipitating com- 
ponents. The diagram may also be used to indicate observed particulate char- 
acteristics such as size, morphology, agglomeration, etc. 

The analysis of solubility curves and precipitation boundaries is illustrated 
for the precipitation of silver bromide (mixing aqeuous solutions of KBr and 
AgN0 3 ) in Figure 8.8 (Fiiredi, 1967). The curves are approximated by tangents 
and the values of the slopes, b, of the various parts of the curve determine the 
compositions of the complexes. For example, slope b=— 1 for this 1 — 1 
electrolyte represents the solubility product and slope b = represents the 
complex AgBr (aq). The other portions of the curve represent the com- 
positions of the other complexes AgBrJ"~ '~ and Ag m Br' m_1 ' + where the slope 

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330 Crystallization 

_ -6 - 

-4 - 

t -2 

Complex ^~ 



solubility f 



/ \ 

I Zone 


/ \ 



/ / 
/ / 


pilation / 







/ 1 1 



O -2 

Ag 2 Br + / 


\"<U / 



Ag 3 Br* + / 



/ N 


Ag 4 Br 3 y / 



/ / 

J / 


/ / 


' / 

f\ ' — 




AgBr 3 2 " 

/ AgBr 4 


/ \ j 1 

1 1 

-6-8 2 O 

Log ( BrVmol L~') 

-6 -8 



Figure 8.8. Precipitation-solubility curves for silver bromide in aqueous solution (a) 
smoothed experimental points, (b) curve approximated by tangents. (After Fu'redi, 1967) 

b = ny — (mx/y). In this case only mononuclear complexes were considered, 
i.e. x = y = m = 1. 

The precipitation boundary may or may not coincide with the solubility 
curve since its position in the diagram depends on the time and method of 
detection of the onset of precipitation. In effect, it establishes the metastable 
zone for the given system. If a stable precipitate is formed at low levels of 
supersaturation the precipitation boundary and solubility curve may be 
assumed to be virtually coincident. In cases where they do not coincide, the 
composition of the critical nuclei may be determined from the slopes of the 
precipitation boundary while the compositions of the corresponding bulk 
equilibrium solid phases may be obtained from the solubility curve. Hence, 
comparison of the two curves can yield information as to whether the composi- 
tion of both nuclei and precipitate is the same or if the bulk solid phase is 
formed by a solid-state transformation from a metastable precursor. A useful 
account of the significance of the zones on a precipitation diagram has also 
been given by Nielsen (1979). 

Another type of 'precipitation diagram' has been proposed for assessing 
hydrometallurgical processes involving the selective precipitation of metal 
hydroxides and salts (Monhemius, 1977). Each diagram illustrates the relative 
solubilities of a particular hydroxide or salt of a range of metals and enables 
estimation of theoretical solubilities at any acidic pH. 

The sulphide precipitation diagram in Figure 8.9 is essentially a plot of cation 
activity against sulphide ion activity. pH is also recorded for systems containing 
H2S at atmospheric pressure. 

The use of the diagram is illustrated as follows. Assess the feasibility of 
removing copper from a solution containing 30gL~'Zn and 2gL~'Cu at 
pHO by precipitation with gaseous H2S. If the solution is saturated with H2S 
at atmospheric pressure, the sulphide ion activity log{S 2 ~} at pH will be —21, 

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Industrial techniques and equipment 


Hg 2 *Ag + Cu 

~ 2+-, 3+ 2+ z + 2* 2 + 2+ 2+ 2+ 2 + 

Cu Bi CdPbSnZnCoNi Fe Mn 






-30 -20 

Log(s z "/mol L"') 

Figure 8.9. Sulphide precipitation diagram at 25 °C. (H2S at atmospheric pressure). (After 
Monhemius, 1977) 

at which the equilibrium activity of copper in solution, as either Cu + or Cu , is 
much less than 10~ 5 molar (below the graph), i.e. the copper is virtually 
insoluble. However, at log{S ~} = —21 the solubility of zinc is also low 
(log{Zn 2+ } = —3.5) so much of the zinc would be co-precipitated with the 
copper. To prevent this occurring, the input of H2S could be controlled so that 
the solution always remains undersaturated with respect to it. For example, by 
keeping log{S ~} lower than say —25 the copper should precipitate, as the 
sulphide, and zinc should remain in solution. 

8.1.5 Techniques of precipitation 

Precipitation by direct mixing 

The precipitation of a solid product as the result of the chemical reaction 
between gases and/or liquids is a standard method for the preparation of many 
industrial chemicals. Precipitation occurs because the gaseous or liquid phase 
becomes supersaturated with respect to the solid component. A crude precip- 
itation operation, therefore, can be transformed into a crystallization process 
by careful control of the degree of supersaturation. 

A common method for producing a precipitate is to mix two reactant solu- 
tions together as quickly as possible, but the analysis of this apparently simple 
operation can be exceedingly complex. Precipitation processes, almost by def- 
inition, involve the creation of highly supersaturated systems and the main 
practical difficulty is to maintain reasonably uniform conditions throughout 
the reaction vessel. The choice of the method of mixing the reactants is there- 
fore very important and the aim should be to avoid any accidental development 
of zones of excessive supersaturation. The sequence of reactant mixing can 

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332 Crystallization 

also be of critical importance: A added to B to produce a precipitate C 
can often yield a very different product from that which results from the 
addition of B to A. The development of local pockets of reactants in non- 
stoichiometric ratios, undesirable pH levels, and so on, can have highly detri- 
mental effects. 

Further comments on the complexity of mixing processes during precipita- 
tion operations are made in section 8.1.7. 

Primary nucleation does not necessarily commence as soon as the reactants 
are mixed, unless the level of supersaturation is very high. The mixing stage 
may be followed by an appreciable time lag (induction period) which depends 
on the temperature, supersaturation, efficiency of mixing, state of agitation, 
presence of impurities, and so on, before nuclei appear. Some time after the 
induction period a rapid desupersaturation ensues (see Figure 5.12) during 
which both primary (homogeneous and heterogeneous) and possibly secondary 
nucleation may occur together, but the predominant process at this stage is 
growth of nuclei. Particle agglomeration followed by ripening or other ageing 
processes can lead to subsequent particle coarsening. 

Precipitation from homogeneous solution 

For the purpose of gravimetric analysis, where it is necessary to effect an 
efficient separation of solid from liquid, it is generally accepted that precipita- 
tion should be carried out slowly from dilute solution. However, some sub- 
stances, such as the hydroxides and basic salts of aluminium, iron and tin, 
demand extremely high dilutions and excessively long times for dense particles 
to be produced. The method known as precipitation from homogeneous solu- 
tion (PFHS) allows coarse precipitates to be produced in relatively short times 
(Gordon, Salutsky and Willard, 1959). 

Briefly, the technique of PFHS consists of slowly generating the precipitating 
agent homogeneously within the solution by means of a chemical reaction. 
Undesirable concentration effects are eliminated, a dense granular precipitate is 
formed and co-precipitation is minimized. For example, silver chloride crystals 
can be produced in aqueous solution by the reactions: 

C 3 H 5 C1 + H 2 -> Cr + C3H5OH + H+ 

Cr + Ag+ -» AgCl 

The growth kinetics of this process are reported to be second order and 
surface reaction controlled. The precipitation of silver iodide in ethanol by 
the reaction 

2C 2 H 5 I + 2AgN0 3 + C2H5OH -> 2AgI + (C 2 H 5 ) 2 + HNO3 + C 2 H 5 N0 3 

is reported to be first order and diffusion controlled. Also first order and 
diffusion controlled are the precipitation of barium sulphate by a persulphate- 
thiosulphate reaction in the presence of Ba 2+ : 

S 2 OJ|- + 2S 2 2 f -» 2SQ2- + S 2 0^ 

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Industrial techniques and equipment 333 

and the production of sulphur (as a spherical monodisperse sol) by the decom- 
position of thiosulphate in acid solution: 

4S 2 Of _ -» 3SO^ + 5S 

Other PFHS methods that have been used for the production of crystalline 
precipitates by the controlled generation of the required anions in an appro- 
priate aqueous solution include the hydrolysis of allyl chloride (Cl~), dimethyl 
oxalate (CiO 1 ^), triethylphosphate (PO^ - ), dimethylsulphate (SO4 - ) and 
thioacetamide (S ~). 

PFHS plays a very important role in analytical chemistry. It is also being 
used in studies of co-precipitation and nucleation phenomena because the slow 
controlled precipitation allows a close approach to equilibrium between the 
solid and solution. The applications of PFHS on the industrial scale have so far 
been limited, but it does appear to be a promising technique. Fractional pre- 
cipitation methods are generally improved by PFHS and it has been applied to 
the difficult separation of radium and barium, for the production of carriers for 
radioactive materials and for the preparation of monodisperse suspensions of 
pigments and polishing agents. 

Reviews of precipitation from homogeneous solution have been made by 
Gordon, Salutsky and Willard (1959); Cartwright, Newman and Wilson (1967); 
Matijevic (1994). 


A solution can be made supersaturated, with respect to a given solute, by the 
addition of a substance that reduces the solubility of the solute in the solvent. 
The added substance, which may be a liquid, solid or gas, is generally referred 
to as the 'precipitant' and the operation is known by a variety of terms. In the 
pharmaceutical industry, for example, for the precipitation/crystallization of 
organic substances from water-miscible organic solvents by the controlled 
addition of water to the solution, the term 'watering-out' is commonly used. 
The descriptions 'drowning-out', 'quenching' and 'solventing-out' have also 
been applied to the precipitation of electrolytes from aqueous solution by the 
addition of a water-miscible organic solvent. However, despite the fact that no 
single designation can be appropriate for all cases, the term 'salting-out' will be 
used for convenience throughout this section. 

The properties required of a liquid precipitant are that it be miscible with the 
solvent of the original solution, at least over the ranges of concentration 
encountered, that the solute be relatively insoluble in it, and that the final 
solvent-precipitant mixture, if comprising valuable components, be readily 
separable, e.g. by distillation. 

Salting-out has many advantages. For example, highly concentrated initial 
solutions can be made by dissolving an impure crystalline material in a suitable 
solvent. If the solute is very soluble in the chosen solvent, dissolution may be 
effected at low temperature and this is advantageous for the processing of heat- 
sensitive substances. By choosing a suitable precipitant, a high solute recovery 

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i 2 C 3 0.4 0.5 

Mole fraction of methanol in the mixed solvent 
Figure 8.10. Solubility of some common salts in methanol-water mixtures at 20°C 

yield may be obtained (see Figure 8.10). Furthermore, better purification is 
often obtained than that from a straightforward crystallization operation since 
the mixed mother liquor often retains more of the undesirable impurities than 
does the original solvent. However, salting-out can have the disadvantage of 
needing a separation unit to handle fairly large quantities of mother liquor if 
the solvent and precipitant have to be recovered. 

A slight dilution of the salting-out agent with the system solvent can be 
extremely beneficial in avoiding excessive nucleation in the regions of primary 
contact. This procedure, demonstrated with aqueous acetone used as a precipit- 
ant for potassium sulphate from aqueous solution, appears to offer consider- 
able advantages for industrial processing in cases where very small primary 
crystals need to be avoided (Mullin, Teodossiev and Sohnel, 1989). A general 
model for controlled precipitant dosage rate addition in batch precipitation has 
been proposed by Jones and Teodossiev (1988). 

Another possibility for reducing supersaturation levels, and hence nucleation 
rates, is to use an air-diluted precipitant. This is quite easy to arrange if the 
precipitant is a volatile organic liquid. A practical example of this technique is 
the foam column described by Halasz and Mullin (1987) which was used for the 
controlled precipitation of potash alum crystals from aqueous solution using 
air saturated with 2-propanol (Figure 8.11). 

The use of hydrotropes, substances which possess the property of selectively 
enhancing the aqueous solubility of sparingly-soluble compounds, can afford 
an opportunity for mixture separation. Colonia, Dixit and Tavare (1998) 
describe an example of the use of a hydrotrope for the separation of the 
o- and /^-isomers of chlorobenzoic acid from a 42/58% mixture, a typical ratio 
for the product from an industrial reactor. Sodium butyl monoglycol sulphate, 
used as a 50% aqueous solution, is a suitable hydrotrope in which the ortho- 

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Industrial techniques and equipment 335 

Figure 8.11. Controlled precipitation in a series of foam columns using an air-diluted 
vapour-phase alcohol as the precipitant for potash alum from aqueous solution. 1, alcohol 
evaporator; 2, 3, precipitation columns; 4, heater coil; 5, thermostat; 6, sintered glass 
distributor; 7, 8, cooling coils; 9, compressed air supply; 10, Rotameter; 11, preheater; 
12, manometer; 13-16, thermometers; 17-20, sampling vessels; 21-23, inlet tubes; 24, 
cyclone; 25, vent to atmosphere. (After Halasz and Mullin, 1987) 

isomer has a much higher solubility than the para- and from which crystals of 
> 99% pure para- can be precipitated when water is added in a controlled 

Gases or solids may be used as precipitants so long as they meet the require- 
ment of being soluble in the original solvent and do not react with the solute to 
be precipitated. Ammonia can assist in the production of potassium sulphate 
by the reaction of calcium sulphate and potassium chloride. In water alone the 
yield is low, but in aqueous ammonia it is greatly improved. Hydrazine can act 
in a similar manner (Fernandez-Lozano and Wint, 1979). High pressure CO2 
has been used to precipitate sulphathiazole from solution in ethanol; the pre- 
cipitation rate, crystal size and habit can be controlled by varying the CO2 
pressure (Kitamura et al., 1997). 

An example of the use of a solid precipitant is the addition of sodium chloride 
crystals for the salting-out of organic dyestuffs. The sodium chloride acts through 
the solution phase, i.e. it must dissolve in the water present before it can act as a 
precipitant, although its precise mode of action is probably quite complex. 

Crystalline salts can be added to solutions to precipitate other salts as a 
result, for example, of the formation of a stable salt pair. This behaviour is 
encountered when two solutes, AX and BY, usually without a common ion, 
react in solution and undergo a double decomposition (metathesis): 

AX + BY ^ AY + BX 

The four salts AX, BY, AY and BX constitute a reciprocal salt pair. One of 
these pairs, AX, BY or AY, BX is a stable pair (compatible salts), which can 
coexist in solution, and the other an unstable salt pair (incompatible salts) 
which cannot (section 4.7.2). 

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336 Crystallization 

Garrett (1958) has described the large-scale production of potassium sul- 
phate from solid glaserite ore (3K2SO4 • Na2S04) which is added to an aqueous 
solution of potassium chloride. Conversion to a stable salt pair occurs, the 
Na2SC>4 and KC1 remain in solution and K2SO4 is precipitated. Mazzarotta 
(1989) studied the cooling crystallization of potassium sulphate from aqueous 
solutions of glaserite and reported the effects of traces of calcium on the 
nucleation and growth kinetics. 

Further reference is made to the use of salting-out in crystallization processes 
in section 8.4.6. 

8.1.6 Industrial applications 

It is not possible to attempt a comprehensive survey of the enormous range of 
industrial precipitation practice, but the examples below will serve to indicate 
something of its diversity. A few references are listed for each topic to serve as 
an introduction to relevant literature. 

Precipitated calcium carbonate is widely used industrially, with different 
applications usually demanding different physical and granulometric proper- 
ties. Three different crystallographic forms of CaCCh can be precipitated 
(calcite, aragonite and vaterite) depending on the supersaturation, temperature, 
presence of trace impurities, etc. Kinetic studies have been reported by 
Maruscak, Baker and Bergougnou (1971). Roques and Girou (1974), Schierholz 
and Stevens (1975), Hostomsky and Jones (1991), Wachi and Jones (1991, 
1995) and Kabasci et al. (1996). Reddy (1978) has made a general survey of 
CaC03 precipitation from waste-waters, and a good introduction to the 
geochemical literature on CaCC>3 precipitation is given by Kitano, Okumara 
and Idogaki (1980). There are many reports of the effects of ionic and other 
trace impurities, e.g. Reddy and Nancollas (1976) and Sohnel and Mullin 

Calcium sulphate can crystallize from aqueous solution as the dihydrate 
(gypsum), hemihydrate (a and j3 forms) or the anhydrous salt (anhydrite: 
a, (3 and 'insoluble' forms), many of which can coexist in contact with solution 
in metastable equilibrium. The solubilities of all forms of calcium sulphate 
decrease with increasing temperature (gypsum only above 32 °C) so CaSC^ 
scale commonly forms on heat-transfer surfaces. Precipitation of calcium sul- 
phate at elevated temperatures, a topic of interest in water-cooling towers, 
petroleum drilling operations, evaporative seawater desalination plants, etc., 
has been studied by Nancollas and Gill (1979). Simulations of calcium sulphate 
precipitation as encountered in the phosphoric acid wet process, have included 
studies of nucleation and growth kinetics (e.g. Amin and Larson, 1968; Sikdar, 
Ore and Moore, 1980) and the effects of trace impurities (e.g. Sarig and Mullin, 
1982; Budz, Jones and Mullin, 1986). 

The precipitation of alumina trihydrate (Bayer hydrate) from caustic 
(sodium aluminate) solution is an important step in the production of alumina 
from bauxite by the Bayer process. The reaction 

2NaAlQ 2 + 4H 2 -> A1 2 3 • 3H 2 Q + 2NaOH 

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Industrial techniques and equipment 337 

is normally carried out in the presence of alumina trihydrate seeds to induce the 
formation of the appropriate species. Agglomeration plays an important part in 
the development of the final product. Kinetic studies of Bayer hydrate pre- 
cipitation include those of Misra and White (1971) and Halfon and Kaliaguine 

Magnesium is recovered from seawater and brines on a large scale world- 
wide by the precipitation of its hydroxide. Some processes use lime as the 
precipitant while others use calcined dolomite (dolime) which reacts with 
MgCl2 according to 

CaO • MgO + MgCl 2 + 2H 2 -> 2Mg(OH) 2 + CaCl 2 

thus enhancing the magnesium yield for a given quantity of seawater. Labor- 
atory studies of Mg(OH) 2 precipitation pertinent to the industrial process have 
been described by Phillips, Kolbe and Opperhauser (1977), Sohnel and Maracek 
(1978), Petric and Petric (1980) and Muffin, Murphy, Sohnel and Spoors (1989). 

Photographic emulsions consist of dispersions of silver halide crystals, 
around 1 um in size, in a protective colloid such as gelatin. The light sensitivity 
of the finished photographic emulsion is to a large extent dependent on the 
crystal size distribution. Emulsions of high light sensitivity are based on silver 
bromide containing small percentages of silver iodide. Most commercial photo- 
graphic emulsions are made by adding silver nitrate solution slowly to an 
agitated vessel containing an aqueous solution of gelatin containing excess 
halide. After precipitation of the silver halide, the dispersion may be held at 
a given temperature to permit ripening (Margolis and Gutoff, 1974; Leubner, 
Jagganathan and Wey, 1980). 

Many pigments, such as Prussian blue, iron oxide reds and yellows, and the 
chromate pigments, are manufactured by precipitation processes. The most 
important group of synthetic organic pigments (Abrahart, 1977) are the azo 
group produced by the diazotization of an aromatic amine followed by coup- 
ling the soluble diazo compound with a suitable agent to form an azo linkage. 
This latter step is the precipitation stage: 

Ar— N+=N + Ar'H -► Ar— N=N— Ar 1 + H+ 

where Ar and Ar' represent different aryl groups. The colour of a pigment 
depends on the physical form in which it is precipitated and this, in turn, 
depends on the kinetics of the coupling reaction and the subsequent nucleation 
and growth steps. The important variables include concentration, temperature, 
pH, time and presence of impurities and/or additives. Stringent particle size and 
size distribution limits around 0.2 um are usually specified. The respective 
merits of batch and continuous coupling methods have been discussed by 
Nobbs and Patterson (1977). Gutoff and Swank (1980) report a pilot-scale 
(300 g/h) study in which amorphous spherical dye particles can be obtained 
continuously by precipitation from solution in a water-miscible solvent by the 
addition of water. 

Crystalline sucrose develops interesting physicochemical properties when 
its specific surface exceeds about 2000cm 2 g _1 , which corresponds roughly to 

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338 Crystallization 

a crystal size of around 20 um. Sucrose crystals of this size and smaller are of 
importance for icing sugar and for use as seed material in large-scale sugar- 
boiling practice. Such small crystals are normally produced by comminution, 
e.g. in hammer mills, or in the latter case by ball-milling in the presence of 
a liquid such as isopropanol. Precipitation, however, would appear to offer an 
alternative procedure with potential advantages. Sucrose may be precipitated 
from aqueous solution by the addition of a water-miscible organic liquid. Of 
a wide range of precipitants studied by Kelly and Mak (1975), 2-methoxy 
ethanol gave the smallest discrete crystals (about 5 um) with the best physical 

There is an increasing use of precipitation for the production of colloidal 
nanoparticles in such fields as pharmacy and cosmetics. Plasari, Grisoni and 
Villermaux (1997) reported an extensive study on the production of ethylcellu- 
lose nanoparticles from solution in ethanol by drowning-out with water. The 
most important process parameter controlling the product particle size distri- 
bution was found to be the ethylcellulose concentration in the ethanolic solu- 
tion. Contrary to all expectations, other parameters such as type of agitator and 
speed, solvent:nonsolvent ratio and temperature had practically no influence. 
In addition, the particle size distribution was insensitive to scale-up and 
a distinct double population of single nanoparticles (<100nm) and aggregates 
(> 200 nm) was obtained in all runs. A comparison was made between the 
experimental results and predictions made from classical nucleation theories 
and aggregative growth models. 

Salting-out is commonly used for the recovery of proteins from solution since 
their solubility is considerably reduced by increasing the ionic strength. Ammo- 
nium and sodium sulphates are most commonly used for this purpose. The 
control of pH is of critical importance: a protein molecule may be represented 
as a dipolar ion and pH determines the overall net charge on the molecule. 
A protein exhibits its minimum solubility at the isoelectric point (zero net charge) 
and for a pure protein this can be very sharp, the solubility increasing greatly 
even at 0.5 pH on either side of the minimum. Detailed studies have been 
reported by Hoare (1982) and Nelson and Glatz (1985). Richardson, Hoare 
and Dunnill (1990) have developed a framework for optimizing the design and 
operation of the fractional precipitation of proteins. They illustrate their 
method experimentally by the precipitation of the enzyme alcohol dehydrogen- 
ase from clarified baker's yeast homogenate using saturated ammonium 
sulphate solution, a system that represents a typical industrial microbial protein 
source and precipitating agent. 

Until comparatively recently, the only reported stable inorganic hydrosols 
were primarily sols of elements such as gold, sulphur, selenium, etc. and 
compounds such as silica, lead iodate, silver halides, etc. A considerable 
amount of attention is now being paid, however, to the preparation of mono- 
dispersed hydrous metal oxides, which are chemically considerably more com- 
plex than other crystalline or stoichiometrically well-defined materials and are 
of interest as potential catalysts. Examples include the hydrous oxides of 
chromium and aluminium (spheres) and copper and iron (polyhedra) with 
particle sizes < 1 um. One manufacturing procedure consists of ageing aqueous 

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Industrial techniques and equipment 339 

solutions of metal salts at elevated temperatures under controlled conditions of 
concentration, pH and temperature for several hours in the presence of com- 
plexing anions, notably sulphate or phosphate (Catone and Matijevich, 1974). 
Her, Matijevich and Wilcox (1990) describe a continuous precipitation method 
for yttrium basic carbonate, in a tubular reactor containing static in-line mixing 
elements, in which aqueous solutions of an yttrium salt and urea are brought 
into contact and subsequently aged. 

Methods and equipment 

Although precipitation is a widely used industrial process, it is disappointing, to 
say the least, to find so few informative published reports on its large-scale 
applications. The vast majority of papers in the scientific and technical liter- 
ature refer to laboratory, and very occasionally to small pilot-scale, operation. 
Nevertheless, there is a wealth of published information available although it is 
scattered widely across the boundaries of many disciplines. 

Most industrial precipitation units are simple in construction. The prime aim 
is usually to mix reacting fluids rapidly and to allow them to develop 
a precipitate of the desired physical and chemical characteristics. Batch opera- 
tion is more often favoured for industrial precipitation than is the truly con- 
tinuous mode, although the merits of semi-continuous operation deserve serious 
consideration, i.e. a continuous reactant mixing step from which the product 
stream passes to batch-operated agitated hold-up tanks to permit equilibration, 
phase transformation, ripening, etc. The hold-up tanks may be arranged in a 
sequenced battery or in a continuous cascade system (see section 9.1.2). 

A distinct advantage of batch precipitation, widely acknowledged in the 
pharmaceutical industry, is that the vessel can be thoroughly washed-out at 
the end of each batch to prevent the seeding of the next charge with an 
undesirable phase that might have arisen from transformation, rehydration, 
dehydration, air oxidation, etc., during the batch cycle. Continuous precipita- 
tion systems often undesirably self-seed after some operating time, resulting in 
the need for frequent shut-down and wash-out. 

Seeded precipitations are occasionally encountered, but very large amounts 
of seed material are generally required compared with those normally utilized 
in conventional crystallization processes (section 8.4.5). It is not uncommon, 
for example, to recycle up to 50 per cent of the magma through a loop system in 
a seeded precipitator to provide the seed surface area needed. 

Several assessments of experimental data from continuously operated pre- 
cipitators, utilizing the population balance (section 8.1.1), are of particular 
interest, e.g. the MSMPR studies on calcium carbonate by Baker and Ber- 
gougnou (1974), calcium sulphate hemihydrate in phosphoric acid (Sikdar, Ore 
and Moore, 1980) and silver bromide in aqueous gelatin suspension (Wey, 
Terwilliger and Gingello, 1980). 

Batch precipitation is more complex than continuous precipitation since the 
basic mechanisms of nucleation and growth can change during the batch 
period. The development process is generally further complicated by the inter- 
vention of agglomeration and ripening. The accounts by Margolis and Gutoff 

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(1974) and Tavare, Garside and Chivate (1980), Mersmann and Kind (1988) 
and Sohnel, Chianese and Jones (1991) highlight some of the problems 
involved. The report by Bryson, Glasser and Lutz (1976) is one of the few 
attempts to compare batch and continuous operation: laboratory-scale data 
are presented for the precipitation of ammonium paratungstate from boiling 
aqueous solutions of ammonium tungstate. 

Reports on truly continuous precipitation on an industrial scale are a rarity. 
The study by Stevenson (1964), therefore, is quite interesting not only because 
it reports a pilot-plant study of a continuous precipitation (of ammonium di- 
uranate by reacting uranyl nitrate solution with ammonia) but also because 
a design route for a multistage continuous unit operated with feedback is 
proposed. Regrettably, there does not appear to have been any follow-up to 
this work. 

The two-stage precipitator {Figure 8.12) described by Aoyama, Kawakami 
and Miki (1979) appears to contain features that could serve other applications. 
In the particular process considered, the feed liquor containing 100 kg m -3 of 
NaOH and 40kgm~ 3 of Al 3+ passes into a suspension of aluminium hydroxide 
seeds kept at 60 °C. Comparatively small crystals of aluminium hydroxide are 
suspended in the upper zone A by solution liquor flowing upwards from zone 
B. As the crystals develop, either by growth or by agglomeration, they fall into 
the lower zone B where they continue to grow. An average slurry density of 
about 25 per cent by mass in zone B allowed particles of around 100 um mean 
size to be produced at approximately 0.75 ton h . 

8.1.7 Mixing techniques 

The mixing of a reactant feedstock stream with the contents of a precipitation 
vessel is of critical importance in the control of precipitate development. It is 


Cj Liquor 
1 overflow 


^~ discharge 

Draft tube 

Figure 8.12. A seeded two-stage precipitator for the recovery of aluminium hydroxide. 
{After Aoyama, Kawakami and Miki, 1979) 

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Industrial techniques and equipment 


essential to achieve good overall mixing to smooth-out supersaturation peaks 
in local regions. Both micromixing and macromixing are involved. The former 
is concerned with mixing at or near the molecular level, and is influenced by 
fluid physical properties and local conditions. The latter, which is concerned 
with bulk fluid movement and blending, is influenced by agitator speed, vessel 
geometry, etc. A third scale of mixing, named meso-mixing by Baldyga and 
Bourne (1997, 1999), is the turbulent dispersion of an incoming fresh feedstock 
plume within a precipitator. These three types of mixing are often considered 
independently, but their contributions to the overall effects of mixing on 
the precipitation process are extremely complex (Garside and Tavare, 1985; 
Sohnel and Mullin, 1987; Villermaux, 1988; Rice and Baud, 1990; Marcant and 
David, 1991; Sohnel and Garside, 1992; Mersmann, 1995; Baldyga and Bourne, 

The position of the feedstock entry point(s) can have a great influence on the 
precipitate quality. A few examples of the many choices available are shown in 
Figure 8.13. If two reactant feedstocks A and B are involved, A could be fed on 
or near the surface of reactant B already in the vessel in the so-called 'single-jet' 
mode {Figure 8.13a) or A could be introduced into the intensely agitated zone 
near the impeller blade {Figure 8.13b). The latter procedure often results in the 
production of larger primary crystals since the good mixing keeps local levels of 
supersaturation low at the first point of contact and minimizes the nucleation 
rate. Reactant B could be introduced as a single jet into reactant A charged first 
into the vessel, if required. The sequence of reactant additions often has 
a significant effect on the characteristics of the precipitate produced. 

Alternatively, both reactant streams A and B may be introduced to the vessel 
simultaneously, in the so-called 'double-jet' mode, and again several choices 
emerge. For example, streams A and B could be introduced together near the 
surface {Figure 8.13c), or near the impeller tip (not illustrated). On the other 
hand, the two streams could be premixed before entering the vessel as a single 
jet {Figure 8.13d) at some appropriate point. 

Premixing of feedstocks is often used because it can provide a means of 
exerting some control over the initial supersaturation levels. Impinging jets or 



A + B 


<^ — 





Figure 8.13. Some possibilities for introducing reactant feedstock streams to a precipita- 
tion vessel 

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342 Crystallization 


Figure 8.14. A vortex mixer for premixing feedstocks. (Courtesy of AEA Technology, 

in-line mixers of a variety of designs (e.g. see Figure 8.14) may be used for 
premixing purposes, some of which will allow complete mixing to be achieved 
in a matter of milliseconds (Demyanovich and Bourne, 1989). 

Controlled double -jet precipitation, which attempts to control primary 
nucleation and the subsequent processes of crystal growth, ripening and 
agglomeration, is a technique that has been used for many years in the photo- 
graphic industry for the production of silver halide microcrystals (< 5 (im). It 
can also be used, with suitable adaptation, for the production of other sparingly 
soluble salts. A comprehensive survey of the wide range of related procedures 
that have been used in the past 40 years to produce monodisperse micro- 
particles, together with detailed accounts of experimental studies using varia- 
tions of controlled double-jet precipitation has been made by Stavek et al. 

A precipitation process can be operated under batch, semi-batch or continu- 
ous conditions, each of which will have its own distinct influence on the 
product crystal size distribution in addition to the combined influences of 
feedstock entry positions, variations in the reactant addition rate profile and 
mixing intensity (Tavare and Garside, 1990; Sohnel and Garside, 1992). 

Guidelines for the choice of optimum reactant solution concentrations in 
a precipitation process have been proposed by Lindberg and Rasmuson (2000) 
who showed that larger crystals are produced when the feed concentration is 
kept low in the early stages of the process and then allowed to increase with 
time in a controlled manner. The term, programmed feed concentration is intro- 
duced for this mode of operation in allusion to the method of programmed 
cooling for batch crystallizers described in section 8.4.5. 

CFD modelling 

Considerable advances have been made in recent years, using computational 
fluid dynamics (CFD), towards a better understanding of mixing effects and 
their influence on precipitation processes (Leeuwen, Bruinsma and van Ros- 
malen, 1996; Wei and Garside, 1997; Leeuwen, 1998; Al-Rashed and Jones, 
1999; Zauner and Jones, 2000b). Several commercial and private CFD 
packages are now available to facilitate solution of the relevant mass, momen- 

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Industrial techniques and equipment 343 

turn and energy conservation equations, and many will also incorporate pre- 
cipitation kinetics and population balance modelling. 

Most of the experimental work aimed at verifying CFD modelling proced- 
ures has so far been performed on small-scale laboratory equipment, but 
encouraging progress is being made. For example, Leeuwen (1998) developed 
a three-compartment model which describes in considerable detail the initial 
turbulent mixing of the feed streams with the surrounding bulk fluid in 
a continuous stirred tank reactor. He compared the findings with experimental 
results for BaSC>4 precipitation with jet mixing in a lV 2 L reactor. Zauner and 
Jones (2000b) developed another approach to the scale-up of both continuous 
and semibatch precipitation of calcium oxalate in reactors ranging in size from 
0.3 to 25 L equipped with a variety of agitators. Residence time/feed time, feed 
concentration, feed point position, feed tube diameter, impeller type and stir- 
ring rate were varied. A segregated-feed model was developed, coupling CFD 
and population balance data, which could lead to the reliable scale-up of 
precipitation processes. It is interesting to note that conventional scale-up 
criteria (equal power input per unit mass, equal tip speed, equal stirring rate, 
etc.) were found incapable of predicting the observed effects of mixing condi- 
tions on the particle size distribution. 

At the time of writing, CFD techniques and the interpretation of their results 
are still in their infancy, but with ever increasing computational power becom- 
ing available, substantial advances towards unravelling the complexities of 
mixing processes may confidently be expected in the very near future. 

8.2 Crystallization from melts 

As explained in section 3.1, the term 'melt' strictly refers to a liquid close to its 
freezing point, but in its general industrial application it tends to encompass 
multicomponent liquid mixtures that solidify on cooling. Melt crystallization is 
the common term applied to the controlled cooling crystallization and separa- 
tion of such systems with the objective of producing one or more of the 
components in relatively pure form. 

Melt crystallization is often considered to be commercially attractive since it 
offers the potential for low-energy separation compared with distillation, 
because latent heats of fusion are generally much lower than latent heats of 
vaporization. The best example in Table 8.1 is that of water where there is 
almost a 7 : 1 difference in the two heat quantities. A further advantage of melt 
crystallization over distillation is that it operates at much lower temperatures, 
and this can be very helpful when processing thermally unstable substances. In 
practice, however, the benefits of low-energy separation for industrial melt 
crystallization can be outweighed by operational problems associated with the 
need to separate purified materials from impure residues. Operating costs can 
also escalate if refrigeration, rather than normal cooling water, is required. Tech- 
nical limitations to the theoretical possibilities for melt crystallization have 
been discussed in some detail by Wintermantel and Wellinghoff (1994). 

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Table 8.1. Enthalpies of crystallization and distillation 






Enthalpy of 


Enthalpy of 






kJkg- 1 


kJkg- 1 


















































o-Nitro toluene 





8.2.1 Basic principles 

Not all melts are amenable to separation by crystallization. The phase equilib- 
ria (Chapter 4) will generally decide the feasibility of the process and often give 
guidance to the choice of the basic procedure to be followed. Only a eutectic 
system {Figure 8.15a) will allow the crystallization of a pure component from 
a melt in one step, but a solid solution system (Figure 8.15b) requires a sequence 
of fractionation steps to yield high-purity products. 

The crystallization of a single component (different for feedstocks to the left 
or right of the eutectic point) in one step from a eutectic system has already 


Solid (A + B) 



Liquid / 
+ / 


a + 



Composition (mass fraction of component B) 


Figure 8.15. Some binary solid-liquid phase diagrams encountered in melt crystallization: 
(a) simple eutectic; (b) simple solid solutions; (c) eutectic with limited solid solubility (a and 
(5 are solid solutions) 

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Industrial techniques and equipment 345 

been discussed in section 4.3.1 {Figure 4.4). The repeated melting and freezing 
steps necessary to produce crystalline products of increasing purity from a solid 
solution system are described in section 4.3.3 (Figure 4.7a). Figure 8.15c shows 
a binary eutectic system with limited solid solubility, which would prevent 
a one-step crystallization from producing a pure component. The existence of 
this type of system should be suspected if product contamination persists in 
a sequence of melt crystallization steps. 

Other more complex systems may be encountered, however, including min- 
imum melting solid solutions, eutectics with compound formation, etc., but in a 
comprehensive survey of binary organic mixtures, Matsuoka (1991) estimated 
that over 50% exhibited simple eutectic behaviour, about 25% formed inter- 
molecular compounds and about 10% formed solid solutions of one kind or 
another. Interestingly, fewer than 2% formed simple solid solutions (Figure 

Ternary eutectics 

The Janecke projection (section 4.4) in Figure 8.16 shows the phase equilibria 
for the ternary system, ortho-, meta- and />ara-nitrotoluene; the three pure 
components are represented by the letters O, M and P, respectively, at the 
apexes of the triangle. Four different eutectics can exist in this system, three 
binaries and one ternary: 

Eutectic points 

Per cent by mass 
































Figure 8.16 may be used to assess a cooling process for mononitrotoluene 
(Coulson and Warner, 1949). For example, if the crystallizer is supplied with 
a liquid feedstock containing 3.0% ortho, 8.5% meta and 88.5% para, it is 
a simple matter to estimate the potential yield of />-nitrotoluene, choosing an 
operating temperature of say 20 °C, as follows. 

Point X in Figure 8.16 represents the composition of feedstock located 
between the 40 and 50 °C isotherms. By interpolation, the temperature at which 
this system starts to freeze can be estimated as about 46 °C. As point X lies in 
the region PBDC, pure para- will crystallize out once the temperature falls 
below 46 °C, and the composition of the mother liquor will follow line XYZ 
(i.e. away from point P) as cooling proceeds. At point Z (about — 15°C) on 
curve DC, meta- also starts crystallizing out. It is not necessary, however, to 
cool to near — 15 °C in order to get a high yield of para-, as the following data, 
based on 100 kg of feedstock, show: 

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Temperature, Para- Mother Composition of mother liquor 

°C deposited, liquor, (mass per cent) 

kg kg O M P 







39.6 60.4 





66.7 33.3 





75.0 25.0 





79.6 20.4 




82.3 17.7 





84.8 15.2 








(-4.1 °C) 


Figure 8.16. Phase diagram for the ternary system o, m- and p-nitroto!uene (point 
D = -40.0°C) 

The above mother liquor compositions are read off Figure 8.16 at the point at 
which the line P — > Z cuts the particular isotherm. The total mass of para- 
crystallized out is calculated by the mixture rule. For example, for 100 kg of 
original mixture X at 20 °C (point Y) 


/distance XY\ 

•deposited = 100 — — - = 75 k 

V distance PY / 

which is equivalent to a recovery of about 85 %. 
8.2.2 Processes and equipment 

Simple agitated vessels, such as those commonly used in solution crystallization 
(section 8.4), rarely find application in melt crystallization processes. One of the 

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Industrial techniques and equipment 347 

main reasons is the difficulty in maintaining efficient mixing in the high-density 
magmas normally encountered; it is not uncommon for more than 90% of 
a melt to crystallize compared with less than 30% of a solution. Massive encrusta- 
tion on the heat-transfer surfaces is another common problem in melt crystal- 
lization, on account of the large temperature driving forces generally used. 

The possibility of overcoming encrustation problems by applying direct 
contact cooling techniques (section 8.4.1) has been explored for the separation 
of organic eutectic systems by suspension melt crystallization (Kim and Mers- 
mann, 1997; Bartosch and Mersmann, 1999). These laboratory studies, which 
utilized both gas (air) and liquid (water) as coolants were aimed at measuring 
heat transfer and kinetic data and observing the crystal product characteristics. 
Eventual industrial application (section 8.2.3) appears possible. 

Sweat-tank crystallizers 

Batch-operated, non-agitated sweat-tank crystallizers were first developed 
towards the end of the 19th century for the processing of phenolic feedstocks 
in the coal-tar industry. A batch of melt is charged into a tank virtually filled 
with coils or fin-tubes that can carry either cooling water or steam. The batch is 
cooled until it almost completely solidifies: crystals first adhere to the cooling 
surfaces. In the second part of the cycle, steam is introduced to the heat transfer 
elements and the adhering crystalline mass begins to melt, impurity inclusions 
are sweated-out and the resulting melt washes away adhering impurities on the 
crystals. The run-off melt is continuously analysed, and when it reaches 
the required product purity, it is directed to a product receiver as the rest of the 
charge is melted. 

These principles were embodied in the Proabd refiner (Societe Proabd, 1959; 
Molinari, 1967a) which also incorporated a feature to prevent the melt run-off 
outlet in the bottom of the tank becoming blocked with crystal during the 
cooling cycle. One proposed method was to fill the bottom of the tank, up to 
the first set of cooling coils, with an immiscible, high-density, low-melting fluid 
which is first run off when the melting process is commenced. For naphthalene, 
for example, a suitable fluid would be a concentrated solution of sodium 

The Sulzer static melt crystallizer with close-packed cooling plates (Figure 
8.17) is a more recent refinement of the sweat-tank concept in which the 
freezing and subsequent melting sequences are time-controlled, the heat transfer 
fluid temperature being lowered and raised according to a pre-set programme. 
Production rates up to 20 000 ton/year and product purities up to 98% are 

Scraped-surface chillers 

A robust mechanical unit that can cope with most encrustation problems is the 
scraped-surface heat exchanger, often referred to as a 'scraped-surface chiller' 
when used in crystallization operations. The unit is essentially a double-pipe 
heat exchanger fitted with an internal scraping device to keep the heat transfer 


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348 Crystallization 

Figure 8.17. The Sulzer static melt crystallizer 

surface clean. The annulus between the two concentric tubes contains the 
cooling fluid, which moves countercurrently to the crystallizing solution flow- 
ing in the central pipe. Located in the central pipe is a shaft upon which scraper 
blades, generally spring-loaded, are fixed. The units range in internal diameter 
from about 150-500 mm and in length up to about 3 m. The close-clearance 
scraper blades rotate at a relatively slow speed (10-60 rev/min) and, operating 
with a AT > 15 °C, an overall heat transfer coefficient in the range 0.2- 
1 kWm~ 2 K _1 may be expected (Armstrong, 1969). 

It should be noted, incidentally, that scraped-surface heat exchangers used 
with homogeneous liquid feedstocks and products normally operate at much 
higher rotor speeds (up to 2000 rev/min) than those of crystallization units and 
give overall heat transfer coefficients up to 4kWm~ 2 K _1 (Skelland, 1958). 

Scraped-surface chillers are employed for the crystallization of fats and 
waxes and the processing of viscous materials such as lard, margarine and ice 
cream (Bailey, 1950; Swern, 1979; Timms, 1991). They are also used for the 
freeze-concentration of foodstuffs such as fruit juices, vinegar, tea and coffee 
(section 8.4.7). The crystals produced by scraped-surface crystallizers are gen- 
erally very small since nucleation and crystal breakage can be excessive. Indeed, 
in some processes, scraped-surface units are often installed for the sole purpose 
of providing nuclei for the growth zone of another crystallizer (see Figures 8.19 
and 8.20). A particular advantage of the scraped-surface crystallizer is that the 
amount of process liquor hold-up is very low. 

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Industrial techniques and equipment 


Column crystallizers 

A process for the fractional crystallization of a melt in a countercurrent column 
was first patented by P. M. Arnold in 1951. In 1961, H. Schildknecht utilized a 
spiral conveyor to transport the solids up or down the column. These tech- 
niques, together with other later refinements, may now be seen in several 
commercial-scale melt crystallizers. 

The principles of column crystallization are shown in Figure 8.18. In the end- 
fed unit (a), a slurry feedstock enters at the top of the column, the crystals fall 
countercurrently to upflowing melt, and there is a heat and mass interchange 
between the solid and liquid phases. When the purified crystals migrate to the 
lower zone, they are remelted to provide high-purity melt for the upflow 
stream. The centre-fed column (b) takes a liquid feedstock. Crystals conveyed 
downwards by means of a screw conveyor are subjected to a surface washing 
action as they come into contact with the counterflowing melt. The high-purity 
crystals reaching the bottom of the column are melted by the heater and 
a liquid product is removed. Some Schildknecht-type columns are operated in 
the reverse mode, i.e. with a heater at the top and a freezer at the bottom, in 
which case the crystals are transported up the column. Although successful as 
a laboratory apparatus (< 200 mm diameter), no large-scale industrial applica- 
tions of the Schildknecht column have yet been reported. 

Models for and analyses of column crystallizer operation have been dis- 
cussed by Bolsaitis (1969); Player (1969); Albertins, Gates and Powers (1967); 
Gates and Powers (1970); Henry and Powers (1970); and Betts and Girling (1971). 

An early example of the commercial application of countercurrent column 
crystallization was the Phillips /^-xylene process (McKay and Goard, 1965; 
McKay, 1967). A concentrated /(-xylene feedstock is cooled from about 10 °C to 
— 18 °C in a scraped-surface chiller cooled with ethylene (Figure 8.19) to pro- 
duce a slurry containing about 40 % of crystals by mass. This slurry feedstock 

slurry feed 

Mother liquor 
(low melting ) 




rLow melting 

■ High melting 



- High melting 



Figure 8.18. Column crystallizers: (a) end-fed column (Arnold type), (b) centre-fed column 
with spiral conveyor (Schildknecht type) 

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65% + 




Scraped surface 

— ir 




5 -I8°C 

■A »- Recycle 




.99 + % 
p- xylene 

Figure 8.19. Phillips countercurrent pulsed column for the production of p-xylene 

passes into a pulsed purification column which acts predominantly as a coun- 
tercurrent wash unit, removing adhering impurities from the crystal surfaces, 
although a small amount of melting and recrystallization does take place. 
A high purity />-xylene melt emerges from the heater section at the base of the 
column. A filter at the top of the column prevents crystals from being carried 
out with the spent mother liquor, which is recycled. An important function of 
the pulse action is to keep the filters clear by inducing a reverse flow through 
the filter medium during part of the pressure pulse cycle. Columns of up to 
70 cm diameter have been operated. 

Brodie purifier 

The Brodie purifier (Brodie, 1971) is a countercurrent melt purification system 
with the added feature of an imposed temperature gradient between the residue 
and product outlets. The unit thereby acts as a countercurrent multistage 
fractionator with partial melting and recrystallization occurring along its length. 
The essential features are shown in Figure 8.20. The heat exchangers, pro- 
vided with slow-moving scraper-conveyors, are arranged into refining and 
recovery sections separated by the feed inlet point. Feedstock enters continu- 
ously and flows through the recovery section towards the cold end (low-melting 
residue outlet) while crystals formed by cooling are conveyed countercurrently 
through the refining section towards the warm end (inlet to the purification 
column). During their passage they are continually subjected to partial melting, 
releasing low-melting impurities which flow back with the liquid stream. The 
interconnected scraped-surface heat exchangers are of progressively smaller 
diameter to maintain reasonably constant axial flow velocities and minimize 
back-mixing. A final purification stage is provided by the vertical column 
purification zone in which the crystals fall countercurrently to an upflow of 
high-purity melt produced from the melter at the base of the column. 

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Industrial techniques and equipment 351 

Residue outlet 
Coolant inlet 

Cooling jackets 

Scraper conveyors-^ 






Scraper conveyor -overflow weir 

^ i; U| 


X Refining 

Section of weir through X-X 


I Product outlet 
Figure 8.20. The Brodie purifier 

Brodie units have been successfully employed for the production of 
high-purity /?-dichlorobenzene and naphthalene at rates up to 7500 ton/year 
(Molinari and Dodgson, 1974). 

TSK process 

The Tsukishima Kikai countercurrent cooling crystallization process (fre- 
quently referred to as the TSK 4C process) is a development of Brodie techno- 
logy with the scraped-surface chillers replaced by several cooling crystallizers 
connected in series. The flow sheet in Figure 8.21 shows three conventional 
agitated cooling crystallizers connected in series. Feed enters the first stage 
vessel and partially crystallizes. The first stage slurry is continuously pumped to 
a hydrocyclone to be concentrated before passing to a Brodie-type column. 
After passing through a settling zone in the crystallizer, clear melt overflows to 
the next stage. Slurry pumping towards the purifying column and clear melt 
overflow from each stage result in a countercurrent flow of liquid and solid 
phases. The process has been applied to the large-scale production of />-xylene 
(Takegami, Nakamaru and Morita, 1984). 

TNO column 

The essential features of the TNO (Toegepast-Natuurwetenschappelijk Onder- 
zoek) column are shown in Figure 8.22 (Arkenbout, 1976, 1978). A crystal 

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Residue v __ j ^ 

1 Product 
Figure 8.21. TSK countercurrent cooling melt crystallization process 

I Coolant 

J I 

Feed ■-£ 

Scraped surface 

Low melting -*- 






m High melting 

Figure 8.22. The TNO vibrating ball and plate column 

slurry is produced in a scraped-surface chiller at the top of the column and 
passes downwards through a series of vibrating sieve trays, each loaded with 
a single layer of metal balls. The action of the balls is to crush the crystals, assist 
their passage through the perforated trays and expose fresh surfaces to the 
countercurrent melt flow. Successful results have been reported with benzene- 
thiophene in 80 mm diameter laboratory columns, and trials have been 

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Industrial techniques and equipment 


reported on a pilot-scale 500 mm diameter column (Arkenbout, 1978). Scale-up 
to commercial size, however, has not yet been achieved. 

Sulzer MWB process 

The Sulzer MWB process (Fischer, Jancic and Saxer, 1984) is a melt crystallizer 
that operates basically by crystallization on a cold surface, but with features 
which allow it to operate effectively as a multistage separation device. Con- 
sequently, it can be used to purify solid solution as well as eutectic systems. 

The effective multistage countercurrent scheme is illustrated for four-stage 
operation in Figure 8.23a. Stage 1, is fed with melt L2 and recycle liquor Li — L, 
where L is the reject impure liquor stream. A quantity of crystals Ci is deposited 
in stage 1. In stage 2 the melted crystals Ci are contacted with melt L3 and fresh 
feedstock F. Subsequent crystallization yields crystals C2 and melt L3. Stages 3 
and 4 follow similar patterns and, in this example, the final high-purity stream 
C4, after being remelted, is split into product C and recycle melt C4 — C. 

Only one crystallization vessel, a vertical multitube heat exchanger, is needed. 
The crystals are not transported; they remain inside the vessel, deposited 
on the internal surfaces of the heat exchanger tubes until they are melted by 


c 2 

L >. 







c 4 

'"l 3 ' 




c 4 -c 


Product loop 

O fl O 

Residue melt 
storage tanks 

Cooling /heating loop 



Heat exchanger 



Residue — CXJ- 

J Coll 


ecting tank 




Figure 8.23. The Sulzer MWB process: (a) multistage flow diagram (C = crystal, L = 
liquor); (h) plant layout (A = crystallizer, B = melt collection tank, C = residue melt 
storage tank, D = heat exchanger, E = pump) 

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354 Crystallization 






Melt bath 



Figure 8.24. Rotary drum crystallizer 

the appropriate incoming warm liquor stream. The control system linking the 
storage tanks and crystallizer consists of a programme timer, actuating valves, 
pumps and the cooling loop (Figure 8.23b). 

The MWB process has achieved remarkable success in the recent years, 
finding large-scale industrial applications in the purification of a wide range 
of organic substances and the separation of fatty acids (Jancic, 1989). Produc- 
tion rates > 100 000 ton/year from groups of linked units are possible. 

Rotary drum crystallizers 

The rotary drum crystallizer (Figure 8.24) is another example of crystallization 
on a chilled surface. A horizontally-mounted cylinder, partially immersed in the 
melt, or supplied with feedstock in some other way, is supplied with coolant 
fluid entering and leaving the inside of the hollow drum through trunnions. As 
the drum rotates, a thin crystalline layer forms on the cold surface and this is 
removed with a scraper knife. 

Agitation of the melt, as near to the drum as possible, appears to have a 
marked effect on the efficiency, as does the drum rotational speed. The opti- 
mum drum speed must be found by trial and error. Decreasing the speed offers 
two advantages: increased contact time which gives a closer approach to 
equilibrium and hence improved separation, and better drainage of impure 
mother liquor off the drum as it emerges from the melt. Decreases in the speed, 
however, reduce the production rate, so a compromise must be reached. 

Rotary drum behaviour and design for melt crystallization have been dis- 
cussed in some detail by Chaty and O'Hern (1964); Chaty (1967); Svalov 
(1970); Toyokura et al. (1976a); Gel'perin and Nosov (1977); and Gel'perin, 
Nosov and Parokonyi (1978), although much of the information relates to 
laboratory, pilot-plant and theoretical studies. Bamforth (1965) describes an 
early use of rotary drum crystallizers for the recovery of sodium sulphate 
decahydrate from rayon spin-bath liquors. 

Continuous belt crystallizer 

The Sandvik continuous cooled belt crystallizer (Figure 8.25) may be considered 
as an alternative to the rotary cooled drum. The underside of the steel belt is 
sprayed with cooling water to provide a controlled temperature gradient along 

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Liquid Product 
Feeding Device 


Industrial techniques and equipment 355 

Solid Product 



w '^' ^' y v v, 

Cooling Water Inlet 4 Outlet 

Figure 8.25. The Sandvik continuous belt crystallizer 

the belt. Melt is fed at one end and crystals are removed at the other, generally in 
the form of thin flakes. The unit can also produce small pastilles if the melt is 
introduced on the belt through a drop-forming feeder. It has also been used for 
wax processing. Belt crystallizers have also been used to produce aluminium 
sulphate flakes and/or pastilles when fed with a highly concentrated seeded 
aqueous solution which rapidly solidifies on contact with the cold surface. 

8.2.3 Alternative procedures 

Most melt crystallizations normally involve, as a basic step, feeding melt onto a 
cold surface where it solidifies. Many variations of this procedure have been 
described above. There are, however, several alternative procedures that merit 

Direct contact cooling 

In this process a melt is cooled and crystallized by direct contact with an inert 
coolant (gas or liquid) which maintains the crystals in suspension in the crystal- 
lizer. This technique has already been exploited in the desalination of sea-water 
by freezing (section 8.4.1 and 8.4.7) and is also a well-established technique in 
the crystallization of inorganic salts from solution (section 8.4.1). It is only in 
recent years, however, that the application to organic melts has been explored, 
but so far only laboratory scale results have been reported. One example of 
such a study is described by Kim and Mersmann (1997) who examined the 
separation of n-decanol/n-dodecanol, a simple eutectic mixture, using air as the 
gaseous coolant (Figure 8.26) and water and liquefied butane as the liquid 
coolants. Bartosch and Mersmann (1999) extended this study on the same 
eutectic mixture with a view to providing design information. Good separations 
were achieved with air at atmospheric pressure, but they identified a potential 
problem with the liquid-solid separation step which could adversely influence 
the product purity. 

Adiabatic evaporative cooling 

An early report of adiabatic evaporative cooling (Takegami, 1993) considered 
the purification of caprolactam (melting point 69 °C) in the presence of water 

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Gas vent 

Figure 8.26. Apparatus for melt DCC crystallization, using air as a coolant. A = 
Reservoir; B = Data acquisition system; C = Thermostatic baths and circulators; D = 
Particle analysis sensor; E= Dilution tank; F '— Temperature recorder; G = Crystallizer; 
H= Flowmeter; I = Refrigerator; J '= Temperature programming controller; K=Heat 
exchanger. (After Kim and Mersmann, 1997) 

which was evaporated under reduced pressure. The process was also considered 
for application to acrylic acid (m.p. 14 °C) and biphenol (m.p. 157 °C). A more 
recent account of the theoretical considerations leading to the selection of 
optimum conditions for this procedure has been given by Diepen, Bruinsma and 
van Rosmalen (2000). Caprolactam was crystallized from water mixtures by 
evaporating of water under reduced pressure; the actual cooling takes place in 
a condenser without any solid phase being present. The optimum process con- 
ditions were found to be 5% water, a pressure of 42mbar and the temperature 


The name 'prilling' is given to a melt-spray crystallization process that results in 
the formation of solid spherical granules. It is employed widely in the manu- 
facture of fertilizer chemicals such as ammonium nitrate, potassium nitrate, 
sodium nitrate and urea. 

In the ammonium nitrate prilling process (Shearon and Dunwoody, 1953) 
a very concentrated solution, containing about 5% water, is sprayed at 140 °C 
into the top of a 30 m high, 6 m diameter tower in which the droplets fall 
countercurrently to an up-flowing air stream that enters the base of the tower 
at 25 °C. The droplets, suddenly chilled when they meet the air stream, solidify 
into 0.5-2 mm diameter prills which are removed from the bottom of the tower 
at 80 °C. As they still contain about 4% water, they are dried at a temperature 
not exceeding 80 °C to ensure that no phase transition occurs (see section 1.8). 
They may then be dusted with diatomaceous earth or some other coating agent 
to prevent caking (section 9.6). The prills made by this process are reported to 

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Industrial techniques and equipment 


have twice the crushing strength of those formed by the spray drying of fused 
water-free ammonium nitrate. 

An interesting method of prilling has been developed for the processing of 
calcium nitrate in a form suitable for use as a fertilizer (van den Berg and 
Hallie, 1960). Calcium nitrate, which is hygroscopic, is produced as a by- 
product in the manufacture of nitrophosphate fertilizer. The process consists 
of crystallizing droplets of calcium nitrate in the form of prills in a mineral oil 
to which seed crystals have been added. A spray of droplets of the concentrated 
solution, formed by allowing jets of the liquid at 140 °C to fall on to a rotating 
cup, fall into an oil-bath kept at 50-80 °C. The prills are removed, centrifuged 
to remove surplus oil, and packed into bags. Because of the thin film of oil 
which remains on the particles, the material is less hygroscopic than it would 
normally be, there is no dust problem on handling, and the tendency to cake on 
storage is minimized. 

In another technique, molten urea is sprayed at 148 °C onto cascading seed 
granules (<0.5mm) in a rotating drum until they grow into product sized 
particles (2-3 mm). Heat released by the solidifying melt is removed primarily 
by evaporation of a fine mist of water sprayed into air that is passed through 
the granulation drum (Shirley, Nunelly and Cartney, 1982). 

Problems associated with the design of prilling towers have been discussed by 
Roberts and Shah (1975); Schweizer, Corelli and Widmer (1975); and Scheffler 
and Henning (1979). The merits of prilling have been compared with granula- 
tion by Ruskam (1978). 

High pressure crystallization 

Pressure, as well as temperature, can alter the equilibrium relationships in a 
melt-crystal system. Figure 8.27a shows the relationship between pressure and 
volume in a liquid crystallizing under the influence of pressure. Curve AB (the 
liquidus) shows the volume decrease as the liquid is compressed isothermally. 
With a pure liquid, crystallization commences at some point B and proceeds at 


\ \ 


Sohdus \ N v 


\ v ^ r' 



c b"\ 

k Liquidus 

c /oc 

/ / b S 


/ h / 
/ / i / 





/ / Y 

/ \s* Liquid: 



Figure 8.27. High pressure crystallization: (a) compression curve showing a solid-liquid 
transition, (b) pressure-temperature relationship in an adiahatic compression 

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358 Crystallization 

constant pressure until all the liquid is solidified (point C). Beyond this point, 
any solid phase compression requires a sharp pressure increase along curve CD 
(the solidus). If the liquid is impure, crystallization will not normally occur 
until some higher pressure than B is applied, e.g. at some point B'. As the 
major component crystallizes, the concentration of the impurity species in 
the remaining liquid increases and, consequently, higher and higher pressures 
are required to continue crystallization, following an exponential path from B' 
to D where the curve approaches the solidus and crystallization is virtually 

The pressure difference between the liquid-solid equilibrium curves for pure 
and impure materials corresponds to the freezing temperature depression 
caused by the presence of an impurity. The pressure increase accompanying 
the increase in pure crystalline material deposited is related to the concentration 
of impurities in the liquid. With this information, therefore, the mother liquor 
composition at any stage can be estimated from the system pressure. Con- 
sequently, a highly pure solid phase can be obtained by separating the liquid 
phase from the solid phase while maintaining an appropriate pressure corres- 
ponding to the desired solid fraction or mother liquor composition. 

Figure 8.27b shows liquid-solid equilibrium lines in terms of pressure and 
temperature. Line a represents the pure component, line b represents the impure 
feedstock and line c represents the eutectic. The sequences of a pressure crystal- 
lization operation are as follows. A liquid feedstock is adiabatically compressed 
from point A to point B and this is accompanied by compression heat genera- 
tion. On further increasing the pressure up to point C, slightly below the eutectic 
line, nucleation occurs and crystal growth proceeds. The temperature rise on 
this occasion is caused by both compression and latent heat release. In this step, 
the impurities are concentrated in the mother liquor which is then separated 
from the solid phase by releasing it from the pressure vessel. When most of the 
mother liquor is discharged, its pressure decreases to atmospheric, while the 
crystals, maintained at the initial separation pressure, are purified by 'sweating'. 
After separation, at point D, the purified crystals gradually approach the 
equilibrium state along line a at point E (atmospheric pressure). 

Amongst the successful separations so far reported (Moritoki, 1984), using 
pressures up to 3000 bar, are those of />-xylene from m—p mixtures, /?-cresol 
from m—p mixtures, and mesitylene from its isomers. 

8.3 Sublimation 

So far, in the discussion of industrial crystallization processes, only the 
crystallization of a solid phase from a supersaturated or supercooled liquid 
phase has been considered. However, the crystallization of a solid substance 
can be induced from a supersaturated vapour by the process generally known 
as 'sublimation'. Strictly speaking, of course, the term sublimation refers only 
to the phase change solid — > vapour without the intervention of the liquid 
phase. In its industrial application, however, the term is commonly used to 
include the condensation (crystallization) process as well, i.e. solid — > vapour 

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Industrial techniques and equipment 359 

— > solid, even though the second step should more properly be referred to as 

In practice, for heat transfer reasons, it is often desirable to vaporize the 
substance from the liquid state, so the complete series of phase changes in an 
industrial sublimation process can be solid — > liquid — > vapour — > solid. It is on 
the condensation side of the process that the appearance of the liquid phase is 
prohibited. The supersaturated vapour must condense directly to the crystalline 
solid state. 

Organic compounds that can be purified by sublimation include: 

2-aminophenol naphthalene 

anthracene 2-naphthol 

anthranilic acid phthalic anhydride 

anthraquinone phthalimide 

benzanthrone pyrogallol 

benzoic acid salicylic acid 

1 ,4-benzoquinone terephthalic acid 

camphor thymol 

Inorganic compounds and elemental substances include: 

aluminum chloride magnesium 

arsenic molybdenum trioxide 

arsenic (III) oxide sulphur 

calcium titanium tetrachloride 

chromium (III) chloride uranium hexafluoride 

iodine water (ice) 

iron (III) chloride zirconium tetrachloride 

The sublimation of ice is an important operation in the freeze-drying of foods 
and biological products. 

Reviews of the industrial applications of sublimation techniques have been 
made by Kemp (1958); Holden and Bryant (1969); Mellor (1978); Kudela and 
Sampson (1986). The basic principles of vaporization and condensation have 
been discussed by Rutner, Goldfinger and Hirth (1964); Strickland-Constable 
(1968); and Mellor (1978). 

8.3.1 Basic principles 

The mechanism of a sublimation process can be described with reference to the 
pressure-temperature phase diagram in Figure 8.28. The significance of the 
P-T diagram applied to one-component systems has already been discussed in 
section 4.2. The phase diagram is divided into three regions, solid, liquid and 
vapour, by the sublimation, vaporization and fusion curves. These three curves 
intersect at the triple point T. The position of the triple point in the diagram is 
of the utmost importance: if it occurs at a pressure above atmospheric, the solid 
cannot melt under normal atmospheric conditions, and true sublimation, i.e. 
solid — > vapour, is easy to achieve. The triple point for carbon dioxide, for 

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360 Crystallization 

1 Liquid 










Figure 8.28. True and pseudo-sublimation cycles 

example, is -57 °C and 5 bar, so liquid CO2 does not form when solid CO2 is 
heated at atmospheric pressure (1 bar); the solid simply vaporizes. If the triple 
point occurs at a pressure less than atmospheric, however, certain precautions 
are necessary if the phase changes solid — > vapour (sublimation) or vapour — > 
solid (desublimation) are to be controlled. For example, because the triple point 
for water is 0.01 °C and 6mbar, ice melts when it is heated above 0°C at 
atmospheric pressure. For ice to sublime, it is necessary to keep both the 
temperature and pressure below the triple point values. 

For industrial processing, it is not uncommon for liquefaction to be allowed 
in the vaporization stage, to facilitate better heat transfer, but this must never 
be allowed in the desublimation (crystallization) step. The condensation equip- 
ment, therefore, must operate well below the triple point. If liquefaction is 
employed before vaporization, the operation is often called pseudo-sublimation. 

Both true and pseudo-sublimation cycles are depicted in Figure 8.28. For the 
case of a substance with a triple point at a pressure greater than atmospheric, 
true sublimation occurs. The complete cycle is given by path ABCDE. The 
original solid A is heated to some temperature represented by point B. The 
increase in the vapour pressure of the substance is traced along the sublimation 
curve from A to B. The condensation side of the process is represented by the 
broken line BCDE. As the vapour passes out of the vaporizer into the con- 
denser, it may cool slightly, and it may become diluted as it mixes with some 
inert gas such as air. Point C, therefore, representing a temperature and partial 
pressure slightly lower than point B, can be taken as the condition at the inlet to 
the condenser. After entering the condenser the vapour mixes with more inert 
gas, and the partial pressure of the substance and its temperature will drop to 
some point D. Thereafter the vapour cools essentially at constant pressure to 
the conditions represented by point E, the temperature of the condenser. 

When the triple point of the substance occurs at a pressure less than atmo- 
spheric, the heating of the solid may easily result in its temperature and vapour 
pressure exceeding the triple point conditions. The solid will then melt in the 
vaporizer: path A to B' in Figure 8.28 represents such a process. However, great 


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Industrial techniques and equipment 


care must be taken in the condensation stage; the partial pressure of the 
substance in the vapour stream entering the condenser must be reduced below 
the triple point pressure to prevent initial condensation to a liquid. The 
required partial pressure reduction can be brought about by diluting the 
vapours with an inert gas, but the frictional pressure drop in the vapour lines 
is generally sufficient in itself. Point C represents the conditions at the point of 
entry into the condenser, and the condensation path is represented by C'DE. 


The separation of two or more sublimable substances by fractional sublimation 
is theoretically possible if they form true solid solutions. The phase diagram for 
a binary solid solution system, at a pressure below the triple point pressures of 
the two components, is shown in Figure 8.29a. Points A and B represent the 
equilibrium sublimation temperatures of pure components A (the more volatile) 
and B, respectively, at the given pressure. The lower curve represents the 
sublimation temperatures of mixtures of A and B while the upper curve repres- 
ents the solid-phase condensation temperatures, often called 'snow points'. 
From Figure 8.29b it can be seen that if a solid solution (e.g. point S) were to 
be partially sublimed (e.g. heated to some temperature X), the resulting vapour 
phase (point Y) would be enriched in component A and residual solid (point Z) 
would be depleted. The mass proportion of vapour to remaining solid is given 
by the ratio of the distance XZ/YX (the mixture rule - see section 4.3.1). 
A repeated procedure of vapour condensation and solid vaporization gives 
the possibility of fractionation, although the practical difficulties in operating 
such a separation process may be considerable. 

Experimental studies on fractional sublimation have been described by Gillot 
and Goldberger (1969); Vitovec, Smolik and Kugler (1978); Matsuoka (1984); 
and Eggers et al. (1986). 


Snow -point 

Composition (mass fraction of component B) 
(a) (b) 

Figure 8.29. (a) Phase diagram for a two-component solid-solution system at a pressure 
below the triple points of the two components A and B; (b) fractional sublimation 

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362 Crystallization 

Vaporization and condensation 

The maximum theoretical rate of vaporization V (kgm~ 2 s _1 ) from the surface 
of a pure liquid or solid under its own vapour pressure is given by the Hertz 
Knudsen equation, which can be derived from the kinetic theory of gases: 

V = P s (M/2ttRT s ) 1/2 (8.17) 

P s is the vapour pressure (Pa) at the surface temperature T S (K), M is the molar 
mass (kgkmoF 1 ) and R the gas constant (8.314 x 10 3 Jkmol -1 K _1 ). 

In practice, however, the actual vaporization rate may be less than that 
predicted by equation 8.17 and it is conventional to include a correction factor 
a (S 1)> generally referred to as an 'evaporation coefficient': 

V = aP s (M /2ttRT s ) 1/2 (8.18) 

A laboratory technique suitable for measuring values of a for sublimable solid 
materials is described by Sherwood and Johannes (1962). 

Rates of sublimation of pure solids into turbulent air streams have been 
successfully correlated (Plewes and Klassen, 1991) by the Gilliland-Sherwood 

d/x = 0.023 Re 03S Sc 0M (8.19) 

where d is a characteristic dimension of the vaporization chamber, x the 
effective film thickness at the vapour-solid interface, and Re and Sc the 
dimensionless Reynolds and Schmidt numbers, respectively. 

Desublimation is generally a transient operation, with the processes of 
simultaneous heat and mass transfer additionally complicated by the effects 
of spontaneous condensation in the bulk gaseous phase (Ueda and Takashima, 
1977). Several steps, which may or may not be independent of one another, can 
be involved in the condensation of a solid phase from a vapour. The first step, 
after the creation of supersaturation in the vapour phase, is nucleation, which 
may be homogeneous but under most circumstances is probably predominantly 
heterogeneous. This event is then followed by both crystal growth and agglom- 
eration in the formation of the final crystal product. A simple laboratory 
technique for making kinetic measurements in subliming systems was described 
by Strickland-Constable (1968) who compared the solid evaporation and 
growth rates of benzophenone under comparable conditions. 

In practice, the two most common ways of creating the necessary super- 
saturation for crystal nucleation and subsequent growth are by cooling through 
a metal surface (which can lead to either a glassy or a multicrystalline deposit, 
both of which necessitate mechanical removal) or by dilution (e.g. with an inert 
gas) which, under suitable conditions, can produce an easily handled loose 
crystalline mass. 

A study of the condensation of several sublimable materials in a fluidized bed 
was reported by Ciborowski and Wronski (1962) and a summary of a study on 
heat and mass transfer processes in a fluidized bed desublimation unit was 
reported by Knuth and Weinspach (1976). The measurement and correlation of 

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Industrial techniques and equipment 363 

condensation heat transfer rates in a pilot plant connected to an industrial 
phthalic anhydride desublimation unit has been described by Bilik and Krup- 
iczka (1983). Krupiczka and Pyschny (1990) have proposed a mathematical 
model for desublimation to assist the selection of the optimum cooling condi- 
tions for smooth and fin-tube heat exchangers. Comparisons are made with 
industrial-scale data for phthalic anhydride. 

8.3.2 Processes and equipment 

Sublimation techniques can be classified conveniently into three basic types: 
simple, vacuum and entrainer. 

In simple sublimation the solid material is heated and vaporized, and the 
vapours diffuse towards the condenser. The driving force for diffusion is the 
partial pressure difference between the vaporizing and the condensing surfaces. 
The vapour path between vaporizer and condenser should be as short as 
possible to reduce the resistance to flow. Simple sublimation has been practised 
for centuries; ammonium chloride, iodine and 'flowers' of sulphur have all been 
sublimed in this manner, often in the crudest of equipment. 

Vacuum sublimation is a natural follow-on from simple sublimation. The 
transfer of vapour from the vaporizer to the condenser is enhanced by reducing 
the pressure in the condenser, which thus increases the partial pressure driving 
force. Iodine, pyrogallol and many metals have been purified by this type of 
process. The exit gases from the condenser usually pass through a cyclone or 
scrubber to protect the vacuum-raising equipment and to minimize the loss 
of product. 

In entrainer sublimation an inert gas is blown into the vaporization chamber 
of a sublimer to increase the rate of flow of vapours to the condensing equip- 
ment and thus increase the yield. Such a process is known as 'entrainer' or 
'carrier' sublimation. Air is the most commonly used entrainer, but superheated 
steam can be employed for substances that are relatively, insoluble in water, 
e.g. anthracene. When steam is used as the entrainer, the vapours may be cooled 
and condensed by direct contact with a spray of cold water. In this manner an 
efficient recovery of the sublimate is made, but the product is obtained in the 
wet state. 

The use of an entrainer in a sublimation process has many desirable features. 
It enhances the flow of vapours from the sublimer to the condenser, as already 
mentioned: it also provides the heat needed for sublimation, and thus an effi- 
cient means of temperature control is provided. The technique of entrainer 
sublimation, whether by gas flow over static solid particles or through a fluidized 
bed, is ideally suited to continuous operation. 

The purification of salicylic acid provides a good example of the use indus- 
trially of entrainer sublimation. Air may be used as the carrier gas; but as 
salicylic acid can be decarboxylated in hot air, a mixture of air and CO2 is 
often preferred. The process shown in Figure 8.30 is carried out batchwise. A 
5-10 per cent mixture of CO2 in air is recycled through the plant, passing over 
heater coils before over the containers, e.g. bins or trays, holding the impure 
salicylic acid in the vaporizer. The vapours then pass to a series of air-cooled 

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Vaporiser-!- r4-i 

Salicylic -<~~y 

acid y'^ , 

H.oter — p^ te ;" p Condense, 

Y Y 


Figure 8.30. An entrainer sublimation process used for salicylic acid purification 

To exhaust 

- -Condenser 

Mill Feed 

Figure 8.31. A typical continuous sublimation unit 

chambers, where the sublimed salicylic acid is deposited. A trap removes any 
entrained sublimate before the gas stream is returned to the heater. Make-up 
CO2 and air are introduced to the system as required, and the process continues 
until the containers are emptied of all volatile matter. 

A typical example of a continuous sublimation plant is shown in Figure 8.31. 
The impure material is pulverized in a mill, and hot air or any other suitable 
gas mixture blows the fine particles, which readily volatilize, into a series 
of separators, e.g. cyclones, where non-volatile solid impurities are removed; 
a filter may also be fitted in the vapour lines to remove final traces of impurity. 
The vapours then pass to a series of condensers. The exhaust gases can be 
recycled or passed to atmosphere through a cyclone or wet scrubber. 

Product yield 

The yield from an entrainer sublimation process can be estimated as follows. 
The inert gas mass flow rate G and mass rate of sublimation S are related by 




where pa and p$ are the partial pressures of the inert gas and vaporized 
substance, respectively. In the vapour stream, and po and ps are their respective 
vapour densities. The total pressure, P, of the system will be the sum of the 
partial pressures of the components 

P = PG+PS 

so equation 8.20 can be written 

S=G Ps 


\PgJ \P - Ps 


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Industrial techniques and equipment 365 

or, in terms of the molecular weights of inert gas, Mq, and the material being 
sublimed, Ms, 

The theoretical maximum yield from an entrainer sublimation process is the 
difference in the sublimation rates corresponding to the conditions in the 
vaporization and condensation stages, respectively. 

For example, salicylic acid (Ms = 138) is to be purified by entrainer sub- 
limation with air (Mq = 29) at 150 °C. The vapours pass to a series of con- 
densers, the internal temperature and pressure of the last being 40 °C and 1 bar 
(10 5 Pa). The air flow rate is 2000 kg h -1 and the pressure drop between the 
vaporizer and the last condenser is 15mbar. The vapour pressures of salicylic 
acid at 150°C and 40 °C are 14.4 and 0.023 mbar, respectively. Therefore, 
assuming saturated conditions: 

Vaporization stage (P = 1.015bar,/> s = 0.0144 bar): 

S v = 200og)f °-° 144 Ul37kgh-' 
V 29 J V1.015- 0.0144; 6 

Condensation stage (P = 1 bar, p$ = 2.3 x 10~ 5 bar): 

^2000@f 2 - 3xl °' 5 5 Uo.22 k gh-' 
V 29 / V 1 -2-3 x 10- 5 / B 

In this particular example, therefore, the loss from the condenser exit gases is 
only 0.22 kgh~' and the theoretical maximum yield is virtually 137 kg h -1 . This 
maximum yield will only be obtained, however, if the air is saturated with 
salicylic acid vapour at 150 °C, and saturation will only be approached if the air 
and salicylic acid are contacted for a sufficient period of time at the required 
temperature. A fluidized-bed vaporizer may allow these optimum conditions to 
be approached; but if air is simply blown over bins or trays containing the solid, 
saturation will not be reached and the actual rate of sublimation will be less 
than that calculated. In some cases the degree of saturation achieved may be as 
low as 10 per cent of the possible value. 

The calculated loss of product in the condenser exit gases is only a minimum 
value. Any other losses due to solids entrainment will depend on the design of 
the condenser and cannot be calculated theoretically. An efficient exit-gas 
scrubber can, of course, minimize these losses. 

Fractional sublimation 

As mentioned above in section 8.3.1, the separation of two or more sublimable 
substances by fractional sublimation is theoretically possible if the sublimable 
substances form true solid solutions, but there have been no reports yet of 
the large-scale commercial exploitation of fractional sublimation. On the other 
hand, a laboratory-scale process known as thin-film fractional sublimation 
(Gillot and Goldberger, 1969) has been successfully applied to the separation 

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Enriching - 

Stripping - 





' Entrainer 

Figure 8.32. Thin-film fractional sublimation column: I, fractionation column; 2, reflux 
condenser; 3, vaporizer; 4, condenser. (After Gillot and Goldberger, 1969) 

of volatile solid mixtures such as hafnium and zirconium tetrachlorides, 
/>-dibromobenzene and />-bromochlorobenzene, and anthracene and carbazole 
(Figure 8.32). A stream of inert non-volatile solids (e.g. glass beads or sand) is 
fed to the top of a vertical fractionation column to fall countercurrently to the 
uprising supersaturated vapour. Vapour movement is facilitated by an up- 
flowing entrainer gas stream. The temperature of the in-flowing solids is main- 
tained well below the snow-point temperature of the vapour and consequently 
the solids become coated with a thin film (< 10 (im) of sublimate which acts as a 
reflux for the enriching section of the column, above the feed entry point. 

Sublimation equipment 

There are virtually no standard forms of sublimation or desublimation equip- 
ment in common use. Most industrial units, particularly on the condensation 
side of the process, have been developed on an ad hoc basis for a specific 
substance and duty. The most useful source of information on equipment types 
is the patent literature, but this has the severe drawback that it offers no 
evidence that a process has been, or is even capable of being, put into practice. 
A wide variety of vaporization units have been used, or proposed, for large- 
scale operation depending on the manner in which the solid feedstock is to be 
vaporized. For example (Holden and Bryant, 1969): 

1. A bed of dry solids, without entrainer gas. This is the simplest arrangement. 

2. Dry solids suspended in a non-volatile heavy liquid. 

3. Solids suspended in a boiling (entrainer) liquid where the entrainer gas is 
formed in situ. 

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Industrial techniques and equipment 367 

4. Entrainer gas flowing through a fixed bed of solid particles. 

5. Entrainer gas bubbling through a melted feedstock, i.e. vaporization takes 
place above the triple point pressure. 

6. Entrainer gas flowing through a dense phase fluidized bed of the solid particles. 

7. Entrainer gas flowing through a dilute phase of the solid particles, e.g. in a 
transfer-line vaporizer where the solid and gas phases are in concurrent flow, 
or a raining solids unit where the solids and entrainer may be in counter- 
current flow. 

8. Vacuum sublimation for cases where the vapour phase consists essentially of 
the sublimate, i.e., when vaporization takes place below the triple-point 
pressure without the aid of an entrainer gas. 

Sublimer condensers usually take the form of large air-cooled chambers which 
tend to give very low heat transfer coefficients, probably not greater than about 
5-10 Wm~ 2 K~'. This is only to be expected because deposits of sublimate on 
the condenser walls act as an insulator. In addition, vapour velocities within the 
chambers are generally very low. Quenching of the vapours with cold air in the 
chamber may increase the rate of heat removal, but excessive nucleation is 
likely and the product crystals will be very small. The condenser walls may be 
kept clear of solid by the use of internal scrapers, brushes and other devices. All 
vapour lines in sublimation units should be of a large diameter, adequately 
insulated and, if necessary, provided with trace heating to minimize blockage 
due to the build-up of sublimate. 

One of the main hazards of air-entrainment sublimation is the risk of fire; 
many substances that are considered to be quite safe in their normal state can 
produce explosive mixtures with air. All electrical equipment should be flame- 
proof, and all parts of the plant should be earthed (grounded) efficiently to 
avoid the build-up of static electricity. Vacuum operation after nitrogen pur- 
ging can provide a much safer processing environment. 

The totally enclosed List sublimation unit {Figure 8.33) is provided with self- 
cleaning heat exchange surfaces and operates semi-continuously under reduced 
pressure without the aid of a carrier gas. Accumulated impurities are dis- 
charged from the sublimator periodically. Batch and continuous modifications 
of this unit are available (Schwenk and Raouzeos, 1995) and have been success- 
fully applied industrially for the purification of anthraquinone, dyestuffs inter- 
mediates, metal-organic compounds and pharmaceuticals with production 
rates ranging from 300 to 10 000 ton/year. 

Calculation of the density of deposited layers of sublimate, and of associated 
variables, as an aid towards the optimization of sublimate condenser design, is 
discussed by Wintermantel, Holzknecht and Thoma (1987). The starting point 
of the analysis is the assumption that the growth of sublimate layers is governed 
mainly by heat and mass transfer; the model is based on conditions in the 
diffusion boundary layer. The main process-determining factors (growth rate, 
mass transfer, and gas concentration) are accounted for. The derived theor- 
etical relationship is shown to fit experimental data. 

A variant on the large-chamber desublimation condenser is a crystallization 
chamber fitted with gas-permeable walls. Vapour and carrier gas is cooled by 


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368 Crystallization 

Crude feedstock 

SH 23 

/ hxi-» Vacuum 


(ij/r AAAA/^ <l - 

J r i ci 







N 2 

Solid impurities 


Pure sublimate 

Figure 8.33. The List AG semi-continuous vacuum sublimation unit. {After Schwenk and 
Raouzeos, 1995) 

evaporation of water dispersed in the cooling space formed by porous walls 
through which the inert gas, which protects the internal walls form solid 
deposits, passes into the cooling space. Crystallization takes place in the bulk 
vapour-gas mixture by direct contact with the dispersed water. Vitovec, Smolik 
and Kugler (1978) used such a system for the partial separation of a mixture of 
phthalic anhydride and naphthalene using nitrogen as entrainer. 

Fluidized-bed condensers (desublimation units) have been considered for 
large-scale application although most of the published reports are concerned 
with laboratory-scale investigations (Knuth and Weinspach, 1976). 

8.4 Crystallization from solution 

The large numbers of different industrial solution crystallizers in existence may 
be classified into a few general categories. Terms such as batch or continuous, 
agitated or non-agitated, controlled or uncontrolled, classifying or non-classi- 
fying, circulating liquor or circulating magma, etc., are useful for this purpose, 
but classification of crystallizers according to the method by which supersat- 
uration is achieved is still probably the most widely used method; thus we have 
cooling, evaporating, vacuum, reaction, etc., crystallizers. 

Many of these classes are self-explanatory, but some require definition. For 
example, the term controlled refers to supersaturation control. The term classi- 
fying refers to the production of a selected product size by classification in 
a fluidized bed of crystals. In a circulating-liquor crystallizer the crystals remain 
in the crystallization zone; only the clear mother liquor is circulated, e.g., 
through a heat exchanger. In the circulating-magma crystallizer the crystals 

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Industrial techniques and equipment 369 

and the mother liquor are circulated together. Any given crystallizer may well 
belong to several of the above types, as the following examples will show. 

8.4.1 Cooling crystallizers 

Non-agitated vessels 

The simplest type of cooling crystallizer is the unstirred tank: a hot feedstock 
solution is charged to the open vessel where it is allowed to cool, often for 
several days, predominantly by natural convection. Metallic rods may be 
suspended in the solution so that large crystals can grow on them and reduce 
the amount of product that sinks to the bottom of the crystallizer. The product 
is removed by hand. 

Because cooling is slow, large interlocked crystals are usually obtained and 
retention of mother liquor is unavoidable. As a result, the dried crystals are 
generally impure. Because of the uncontrolled nature of the process, product 
crystals range from a fine dust to large agglomerates. 

Labour costs are generally high, but the method may be economical for small 
batches because capital, operating, and maintenance costs are low. However, 
the productivity of this type of equipment is low and space requirements are high. 

Agitated vessels 

The installation of an agitator in an open-tank crystallizer generally results in 
smaller, more uniform crystals and reduced batch time. The final product tends 
to have a higher purity because less mother liquor is retained by the crystals 
after filtration and more efficient washing is possible. Vertical baffles may be 
fitted inside the vessel to induce better mixing, but they should terminate below 
the liquor level to avoid excessive encrustation. For the same reason, water 
jackets are usually preferred to coils for cooling purposes and, where possible, 
the internal surfaces of the crystallizer should be smooth and crevice-free 
(section 9.5). 

An agitated cooler is more expensive to operate than a simple tank crystal- 
lizer, but it has a much higher productivity. Labour costs for product handling 
may still be rather high. The design of tank crystallizers varies from shallow 
pans to large cylindrical tanks. 

The use of external circulation allows good mixing inside the crystallizer and 
high rates of heat transfer between the liquor and coolant (Figure 8.34a). An 
internal agitator may be installed in the crystallization tank if needed. The 
liquor velocity in the tubes is high; therefore, small temperature differences are 
usually adequate for cooling purposes and encrustation on heat-transfer sur- 
faces can be reduced considerably. The unit shown may be used for batch or 
continuous operation. 

The large agitated cooling crystallizer shown in Figure 8.34b has an upper 
conical section which slows down the upward velocity of liquor and prevents 
the crystalline product from being swept out with the spent liquor. An agitator 
located in the lower region of a draft tube circulates the crystal slurry (magma) 

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Feed Magma 






Magma to 
<^> filter 


Figure 8.34. Agitated tank crystallizers: (a) external circulation through a heat exchanger, 
(h) internal circulation with a draft tube 

through the growth zone of the crystallizer. If required, cooling surfaces may be 
provided inside the crystallizer. Units of this type, called Pachuca growth-type 
crystallizers, have been used for the large-scale production of borax from 
natural brines at Trona, California (Garrett, 1958). 

The batch time of a cooling crystallizer is often prolonged because of the 
build-up of crystal encrustation on the cooling coils, or other heat exchange 
surfaces, resulting in progressively lower heat transfer coefficients. One way of 
overcoming this problem is to carry out the cooling process in two steps (Nyvlt, 
1978). In the first stage, operated with a high temperature difference across the 
heat exchanger, rapid cooling is allowed to proceed until a significant amount 
of encrustation is deposited on the cooling surfaces. At this point, the contents 
of the first vessel are discharged into a second vessel where cooling continues, 
this time under the influence of a lower temperature difference selected so that 
encrustation is avoided. The total batch time of these two stages can be very 
much shorter than that of a conventional previously single-stage operation, and 
the cooling water consumption significantly reduced. A worked example is 
given in section 9.3.3 (example 9.4). 

In practice, if the two vessels are arranged one above the other, the following 
sequence of operations can be adopted. Hot feedstock solution is charged to the 
upper stage 1 crystallizer where it is quickly cooled to the predetermined 
temperature and then discharged into the lower stage 2 vessel to be cooled to 
the final batch temperature before discharging the magma, e.g. to a centrifuge. 
Meanwhile, the upper vessel is charged with fresh hot feedstock, during which 
operation the encrustations that had formed are dissolved. Fast cooling can 
then begin again and the cycle repeated. 

A unit called the 'twin' or 'double' crystallizer is claimed to give a product 
with a very narrow size distribution (Nyvlt, 1971). It consists of two intercon- 

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Industrial techniques and equipment 


Figure 8.35. The 'twin' or 'double' crystallizer 

nected simple crystallizers, each section operating at a different temperature 
(Figure 8.35). Hot feed liquor enters and mixes with the circulating contents of 
the crystallizer, which pass downwards through the water-cooled draft tube, A, 
under the influence of an agitator. Part of the cooled magma passes under the 
adjustable gate, B, into the second compartment of the crystallizer, where it 
mixes with the circulating magma in the second draft tube, C, operated at 
a lower temperature. A back-flow of magma occurs above the gate. Large 
crystals migrate to the bottom of the crystallizer and are discharged. The 
mother liquor exit is located behind the baffle. If both sections of the crystal- 
lizer operate with supersaturated solution, the system functions essentially as 
crystallizers in series. If one section operates slightly above the saturation 
temperature, the excess fines are dissolved and coarse crystals with rounded 
edges are produced. A detailed analysis of the performance of a double crystal- 
lizer is given by Skfivanek et al. (1976). 

Trough crystallizers 

The first truly continuous crystallizer to be introduced to the chemical industry, 
between 1905 and 1910, was the Wulff-Bock unit (Figure 8.36), frequently 
referred to as the crystallizing cradle or rocking crystallizer. It consists of a long 
shallow trough, about 1.2 m wide, rocked on supporting rollers. The solution to 
be crystallized is fed in at one end and the crystals are discharged at the other 
end, continuously. Transverse baffles may be fitted inside the trough to prevent 
longitudinal surging of the liquor, so the charge flows in zigzag fashion along the 
unit. The slope of the trough, towards the discharge end, is varied according to 
the required residence time of the liquor in the crystallizer. 

One of the advantages of the Wulff-Bock crystallizer is the complete absence 
of moving parts in the crystallization zone. Several units may be joined together 
and assemblies up to 30 m in total length have been installed. No external 
cooling is employed; heat is lost by natural convection to the atmosphere. High 
degrees of supersaturation, therefore, are not encountered at any point within 

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Figure 8.36. Wulff-Bock crystallizer. {After Griffiths, 1925) 

the unit and crystallization occurs slowly. The gentle agitation prevents crystal 
attrition and very large (~1 cm) uniform crystals, e.g. of sodium thiosulphate, 
have been grown at production rates of 2-3 ton/day (Griffiths, 1925). Potas- 
sium chloride, potassium permanganate, sodium acetate, sodium sulphate and 
sodium sulphite have also been produced commercially in this manner. Only 
a few Wulff-Bock crystallizers still remain in service. 

The Swenson-Walker crystallizer (Figure 8.37) developed in the early 1930s 
is a well-known example of a trough crystallizer with internal agitation and 

Figure 8.37. Swenson-Walker crystallizer. (After Seavoy and Caldwell, 1940) 

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Industrial techniques and equipment 


a cooling system. Each unit consists of a semi-cylindrical trough up to about 1 m 
wide and 3-5 m long, generally fitted with a water-cooled jacket. Several units 
may be linked together. A helical agitator-conveyor rotates at a slow speed 
(5-10 rev/min) inside the trough to aid the growth of the crystals by lifting them 
and then allowing them to fall back through the solution. The system is kept in 
gentle agitation and the crystals are conveyed along the trough. Overall heat 
transfer coefficients of ~ 50-105 Wm~ 2 K _1 based on a logarithmic mean 
temperature difference between solution and cooling water, may be expected. 
Moderately sized and fairly uniform crystals can be obtained from this type of 
crystallizer and production rates of up to 20 ton/day of salt such as sodium 
phosphate and sodium sulphate have been reported from a single unit (Seavoy 
and Caldwell, 1940). 

Crystallizers similar to the Swenson-Walker type with semi-cylindrical 
(U-type) or nearly cylindrical (O-type) cross-section have been used for many 
years in the sugar industry for the crystallization of concentrated molasses. 
Units fitted with water-cooled jacket are still employed, but these show poor 
heat transfer characteristics; scraper-stirrers are not permitted on account of 
the damage they do to the crystals. Accordingly, many other types of cooling 
arrangement have been tried (Seavoy and Caldwell, 1940). 

Cooling disc crystallizer 

The first cooling disc crystallizer to be developed, in the early 1930s, was the 
Werkspoor 'rapid' crystallizer which has widely used in the sugar industry for 
the processing of after-product massecuite. It was an open trough machine 
containing a horizontally mounted slow-speed agitator-cooler in the form of 
hollow discs through which cooling water was circulated. The discs had seg- 
mental openings to enable the crystal slurry to flow through the machine 
countercurrently to the cooling medium. This crystallizer subsequently found 
another large-scale application in the recovery of Glauber's salt from rayon 
spin-bath liquors (Bamforth, 1965). 

In the modern Gouda cooling disc crystallizer (Figure 8.38) the discs them- 
selves are not heat exchangers. This multistage crystallizer consists of an open 
or closed trough, the latter being designed for the processing of toxic and 
flammable materials. The trough is effectively divided into a number of com- 
partments by fixed vertical plate heat exchangers between which discs with 
segmental openings slowly rotate. Wipers on the discs can help to keep the 
cooling surfaces free from crystalline deposits. The crystal slurry flows from 












(a) (b) 

Figure 8.38. The Gouda MF cooling disc crystallizer (a) open type, (b) closed type 

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374 Crystallization 

one compartment to the next, through bottom openings in the heat exchangers, 
and the temperature decreases stepwise. Applications in a wide range of solu- 
tion crystallizations are claimed. 

Rotary crystallizers 

Rotating cylinders, similar in some respects to those used as rotary driers or 
kilns, have been used for the crystallization of solutions. The cylinder slopes 
slightly from the feed liquor inlet down to the crystal magma outlet. Cooling 
may be provided either by cold air blown through the cylinder or by water 
sprayed over the outside. In the former case internal baffles disturb the liquor 
on the inside wall and cause it to rain through the air stream. In the latter case 
internal scraper devices prevent excessive build-up of crystal on the walls. 

Bamforth (1965) describes several different units used for the recovery of 
ferrous sulphate from pickle liquors and sodium sulphate from spin-bath liquors. 

Internally chilled rotating drum crystallizers (see Figure 8.24) are normally 
associated with melt crystallization (section 8.2), but have also found occa- 
sional application for crystallization from solution. Sodium sulphate and bar- 
ium hydroxide hydrates, for example, have been produced commercially in this 

Scraped-surface crystallizers 

High heat transfer coefficients, up to 1 kW m~ 2 K _1 , and hence high production 
rates, are obtainable with double-pipe, scraped-surface heat exchangers. 
Although mainly employed in the crystallization of fats, waxes and other 
organic melts (section 8.2.2) and in freeze concentration processes (section 
8.4.7), scraped-surface chillers have occasionally been employed for crystal- 
lization from solution. Because of the high turbulence and surface scraping 
action, however, the size of crystal produced is extremely small. 

Oslo-Krystal cooling crystallizer 

Towards the end of the First World War, investigations were carried out in 
Norway by Isaachsen and Jeremiassen into the problems associated with the 
continuous production of large uniform crystals, in particular of sodium chlor- 
ide. These investigations subsequently led to the development, by Jeremiassen, 
of a method for maintaining a stable suspension of crystals within the growth 
zone of a crystallizer. The practical application of this method has been 
incorporated in a continuous classifying crystallizer known by the names Oslo 
or Krystal (Jeremiassen and Svanoe, 1932). Bamforth (1965) has given a con- 
cise account of the design and uses of these versatile and compact units. 

There are several basic forms of the Oslo apparatus, but all units based on 
the original Jeremiassen process have one feature in common - a concentrated 
solution, which is continuously cycled through the crystallizer, is supersat- 
urated in one part of the apparatus, and the supersaturated solution is con- 
veyed to another part, where it is gently released into a mass of growing 

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Industrial techniques and equipment 


Crystals s -^ 


Figure 8.39. Oslo-Krystal cooling crystallizer 

crystals. These units, therefore, belong to the 'circulating liquor' type of 

The operation of the Oslo cooling crystallizer (Figure 8.39) may be described 
as follows. A small quantity of warm concentrated feed solution (0.5 to 2 per 
cent of the liquor circulation rate) enters the crystallizer vessel at point A, 
located directly above the inlet to the circulation pipe B. Saturated solution 
from the upper regions of the vessel, together with the small amount of feed 
liquor, is circulated by pump C through the tubes of heat exchanger D, which 
is cooled rapidly by a forced circulation of water or brine. On cooling, the 
solution becomes supersaturated, but not sufficiently for spontaneous nucle- 
ation to occur, i.e. metastable, and great care is taken to prevent it entering the 
labile condition. The temperature difference between the process liquor and 
coolant should not normally exceed 2°C. The supersaturated solution flows 
down pipe E and emerges from the outlet F, located near the bottom of the 
crystallizer vessel, directly into a mass of crystals growing in the vessel. The rate 
of liquor circulation is such that the crystals are maintained in a fluidized state 
in the vessel, and classification occurs by a hindered settling process. Crystals 
that have grown to the required size fall to the bottom of the vessel and are 
discharged from outlet G, continuously or at regular intervals. Any excess fine 
crystals floating near the surface of the solution in the crystallizer vessel are 
removed in a small cyclone separator H, and the clear liquor is introduced back 
into the system through the circulation pipe. A mother liquor overflow pipe is 
located at point I. 

Like all other cooling crystallizers, this unit can only be used to advantage 
when the solute shows an appreciable reduction in solubility with decrease in 
temperature. Examples of some of the salts that can be crystallized in this 
manner are sodium acetate, sodium thiosulphate, saltpetre, silver nitrate, cop- 
per sulphate, magnesium sulphate and nickel sulphate. Bamforth (1965) reports 
the production of 7 ton/day of 10 x 5 mm sodium thiosulphate crystals in a 2 m 
diameter 6 m high vessel with 200 m 2 heat exchange surface. 

Direct contact cooling 

The simplest form of direct contact cooling (DCC) is effected by blowing air 
into a hot crystallizing solution. Cooling takes place predominantly through 

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376 Crystallization 

the evaporation of water and the air flow also serves as a means of agitation. In 
recent years a considerable amount of attention has been given to the possib- 
ility of using other direct contact coolants for crystallization processes. 

By avoiding the use of a conventional heat exchanger, a DCC crystallizer 
avoids the troublesome problems associated with encrustation on heat transfer 
surfaces (section 9.5). The coolant in a DCC process can be a solid, liquid or 
gas, and heat is extracted by the transfer of sensible and/or latent heat (vapor- 
ization or sublimation). 

Four main modes of action can be specified depending on the degree of 
solubility of the coolant in the solution and the manner in which heat is 
transferred. Within each mode of operation further sub-divisions of procedure 
may be possible: 

1. Immiscible I boiling DCC. The coolant, solid or liquid, is substantially inso- 
luble in the solution and its latent heat of phase change (of sublimation or 
vaporization) is the main cause of heat removal. 

2. Immiscible/non-boiling DCC. The coolant, solid, liquid, or gas, is substan- 
tially insoluble in the solution and its sensible heat is utilized for cooling 

3. M is ciblej boiling DCC. The coolant, usually a liquid, is substantially soluble 
in the solution and its latent heat of vaporization is utilized. 

4. Misciblejnon-boiling DCC. The coolant, usually a liquid, is substantially 
soluble in the solution and its sensible heat is utilized. 

Examples of proposed DCC coolants include liquid butane for the seawater 
desalination process (section 8.4.7) and methyl ethyl ketone for the Dilchill 
lubricating oil dewaxing process (Bushnell and Eagen, 1975). Chlorinated 
hydrocarbons, fluorocarbons and CO2 have also found application in specific 

The continuous DCC crystallizer shown in Figure 8.40 has been used for the 
large-scale production of calcium nitrate tetrahydrate (Cerny, 1963). Aqueous 
feedstock enters at the top of the crystallizer at 25 °C and flows countercur- 
rently to the immiscible coolant droplets, e.g. petroleum, introduced into the 
draft tube at — 15 °C. The magma, containing crystals of mean size ^500 um, is 
discharged at — 5°C. The low-density coolant collects in the upper layers and 
passes to a cyclone to separate aqueous solution droplets before being recycled. 

The thermal efficiencies of the Cerny and similar column crystallizers have 
been discussed by Letan (1973) in the light of data obtained for the crystal- 
lization of MgCh • 6H2O from aqueous solution using kerosene as a coolant. 
Shaviv and Letan (1979) studied the effect of operating conditions (temper- 
ature, droplet size and hold-up) on the same process. 

Examples of the use of DCC crystallization for the recovery of valuable 
components from waste liquids are given by Toyokura et al. (1976a) and 
Nagashima and Yamasaki (1979) who report data on pilot plant units using 
CCI2F2 as coolant to recover 5 ton/day of sodium hydroxide. Duncan and 
Phillips (1976) have discussed the potential application of DCC crystal- 
lization to the production of /^-xylene, using CCIF3, liquid nitrogen and liquid 
natural gas as coolants. Mullin and Williams (1984) have compared indirect 

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Industrial techniques and equipment ill 

Feed liquor 25°C 

Layer of 
immiscible coolant 

Rising droplets 
of coolant 

Central tube 


Slow-speed agitator 


Crystal magma 

Figure 8.40. The Cerny direct-coolant crystallizer. (After Bamforth, 1965) 

cooling with DCC, using iso-octane, for the crystallization of potassium 
sulphate. Kim and Mersmann (1997) and Bartosch and Mersmann (1999) 
have explored the possibility of using DCC crystallization to separate indi- 
vidual components from eutectic organic melts (section 8.2.3). 

A general review of DCC crystallization by Casper (1981) outlines a number 
of processes and equipment details. 

8.4.2 Evaporating crystallizers 

When the solubility of a solute in a solvent is not appreciably decreased by a 
reduction in temperature, supersaturation of the solution can be achieved by 
removal of some of the solvent. A number of evaporation techniques are 

Solar evaporation 

The evaporation of brines in shallow ponds, using energy from solar radiation, 
has been practised for thousands of years and still provides an important means 
of recovering salts from saline waters in many parts of the world (Sonnenfeld, 
1984). An excellent example is the recovery of salts from Dead Sea waters 
(Novomeysky, 1936). 

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378 Crystallization 

In principle, the technique is simple, but control of the complex solar evap- 
oration process still presents a considerable number of technical problems 
(Finkelstein, 1983). Bonython (1966) has given a comprehensive analysis of 
the many factors that can determine the rate of evaporation from solar ponds. 
The use of organic dyes or the encouragement of the growth of certain naturally 
coloured microorganisms in the brine to assist the trapping of incoming solar 
radiation has received a lot of attention in recent years. A study to compare the 
relative efficiencies of dyestuffs and suspensions of halophilic bacteria was 
made by Jones, Ewing and Melvin (1981). 

Attempts to model solar pond operations have been made for existing 
commercial productions of Na2CC>3 • IOH2O (Manguo and Schwartz, 1985) 
and KC1 (Klein et ah, 1987). The use of weather-station data in the design 
and operation of solar ponds is discussed by Butts (1993) and correlations for 
solar evaporation rates using weather variables have been proposed by Lukes 
and Lukes (1993). 

Steam-heated evaporators 

Most evaporation units are steam heated and a typical evaporator body used in 
evaporative crystallization is the short-tube vertical type in which steam con- 
denses on the outside of the tubes (Figure 8.41). A steam chest, or calandria, 
with a large central downcomer allows the magma to circulate through the 
tubes; during operation the tops of the tubes are just covered with liquor. To 
increase the rate of heat transfer, especially in dealing with viscous liquors, 
a forced circulation of liquor may be effected by installing an impeller in the 



I 1 — ~ Vapour 



— "-Magma 

Figure 8.41. A typical crystallizing evaporator containing a calandria with a large central 

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S — 



Industrial techniques and equipment 379 

(a) (b) 

Figure 8.42. Principle of multiple-effect evaporation: (a) single-effect; (b) double-effect 

Multiple-effect evaporation 

Low pressure steam, i.e. < 4 bar, is normally used in evaporators, and fre- 
quently by-product steam (~ 1.5-2 bar) from some other process is employed. 
Nevertheless, 1 kg of steam cannot evaporate more than 1 kg of water from 
a liquor, and for very high evaporation duties the use of process steam as the 
sole heat source can be very costly. However, if the vapour from one evapor- 
ator is passed into the steam chest of a second evaporator, a great saving can be 
achieved. This is the principle of the method of operation known as multiple- 
effect evaporation (Figure 8.42). As many as six effects have been used in 

It is beyond the scope of this book to deal in any detail with the subject of 
multiple-effect evaporation; all standard chemical engineering textbooks con- 
tain adequate accounts of this method of operation, but two important points 
may be made here. First, multiple-effect evaporation increases the efficiency of 
steam utilization (kg of water evaporated per kg of steam used) but reduces the 
capacity of the system (kg of water evaporated). The well-known equation for 
heat transfer may be written 


where Q is the rate of heat transfer, U the overall heat transfer coefficient, A the 
area of the heat exchanger, and AT the driving force, the temperature differ- 
ence across the heat transfer septum. The area A is usually fixed, so the 
variables to consider are Q, U and AT. Q will be reduced by heat losses from 
the equipment, U by sluggish liquor movement and scaling in the tubes of the 
calandria, by the increase in the boiling point of the liquor as it gets more and 
more concentrated. These and many other factors prevent the achievement of 
the ideal condition: 

1 effect 

2 effects 

3 effects 

1 kg steam — > 1 kg vapour 
1 kg steam — > 2 kg vapour 
1 kg steam — > 3 kg vapour, etc. 

Nevertheless a close approach to this ideality can often be produced. Some of 
the evaporators in a multiple-effect system are frequently operated under 
reduced pressure to reduce the boiling point of the liquor and thereby increase 
the available AT. 

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380 Crystallization 

Before leaving this brief account of multiple-effect evaporation, mention may 
be made of the various methods of feeding that can be employed. Figure 8.43 
shows, in diagrammatic form, the possible feed arrangements. In all cases fresh 
process steam enters effect number one. S denotes live steam, F the feed 
solution, V vapour passing to the condenser system, and L the thick liquor, 
or crystalline magma, passing to a cooling system or direct to a centrifuge. 
Pumps are indicated on these diagrams to indicate the number required for 
each of the systems. In all cases the vapours flow in the direction of effects 
1 -> 2 -> 3 ->, etc. 

In the forward-feed arrangement (Figure 8.43a) liquor as well as vapour pass 
from effect 1 — ^ 2 — > 3 — >, etc. As the last effect is usually operated under 
reduced pressure, a pump is required to remove the thick liquor; a feed pump 
is also required. The transfer of liquor between the intermediate effects is 
automatic as the pressure decreases in each successive effect. Suitable control 
valves are installed in the liquor lines. The disadvantages of a forward-feed 
arrangement are (a) the feed may enter cold and consequently require a con- 
siderable amount of live steam to heat it to its boiling point; (b) the thick liquor, 
which flows sluggishly, is produced in the last effect, where the available AT is 
lowest; (c) the liquor pipelines can easily become steam-blocked as the liquor 
flashes into the evaporator body. 

In the backward-feed arrangement (Figure 8.43b) the thick liquor is produced 
in effect 1, where the AT is highest; the liquor is more mobile on account of the 
higher operating temperature. Any feed preheating is done in the last effect, 
where low-quality steam (vapour) is being utilized. More pumps will be 
required in backward feeding than in forward feeding; the liquor passes into 
each effect in the direction of increasing pressure. No feed pump is necessary, as 
the last effect is under reduced pressure. Liquor does not flash as it enters an 
evaporator body, so small-bore liquor lines can be used. Backward feeding is 
best for cold feed liquor. 

In the parallel-feed arrangement (Figure 8.43c) one feed pump is required, 
and predetermined flows Fi, F2 and F3 are passed into the corresponding 
effects. A thick liquor product is taken from each effect: for this purpose pumps 
are generally necessary. Parallel feeding is often encountered in crystallization 
practice, e.g. in the salt industry, and it is useful if a concentrated feedstock is 
being processed. 

In Figure 8.43d a 5-effect system is chosen to illustrate the mixed-feed 
arrangement. Many different sequences are possible; the one demonstrated is 
as follows. Feed enters an intermediate effect (number 3 in this case) wherefrom 
the liquor flows in forward-feed arrangement to the last effect, is then pumped 
to effect number 2 and from there flows in backward-feed arrangement to the 
first effect. Some of the advantages of this method of feeding are (a) fewer 
pumps are required compared with backward feeding; (b) the final evaporation 
is effected at the highest operating temperature; and (c) frothing and scaling 
problems are claimed to be minimized. Caustic soda evaporation is often 
carried out on a mixed-feed basis. 

An alternative to multiple-effect evaporation is vapour recompression. The 
principle of the method is to raise the temperature of the vapour leaving the 


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Industrial techniques and equipment 



(a) Forward feed 

S — 


— J 6- 

(b) Backward feed 

6-- L: i 

:F 2 

6— L 2 

5— L 3 

(c) Parallel feed 

L— o Q — - 


F — ■© 

(d) Mixed feed 

Figure 8.43. Various feeding arrangements in multiple-effect evaporators 

evaporator, by raising its pressure, and passing it back into the evaporator 
heat exchanger. The recompression may be effected either mechanically (with 
a compressor) or thermally (with an injection of high pressure steam). An advant- 
age of vapour recompression is that only one evaporator body is needed. 
A disadvantage is that it is not applicable to systems with high boiling point 
elevations. An economic comparison of multi-effect and vapour recompression 
evaporation has been made by Wohlk (1982). King (1984) discusses some of the 

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merits of mechanical vapour recompression and outlines the optimization of 
operation for the production of Na2SC>4 • IOH2O. 

Forced circulation evaporators 

The typical forced circulation evaporator-crystallizer shown in Figure 8.44a is a 
circulating magma unit operated under reduced pressure. Magma is circulated 
from the conical base of the evaporator body through the vertical tubular heat 
exchanger and reintroduced tangentially into the evaporator below the liquor 
level to create a swirling action and prevent flashing. Feedstock enters on the 
pump inlet side of the circulation system. The liquor level in the separator (the 
evaporator body) is kept above the top of the heat exchanger to prevent boiling 
in the tubes. The fine crystals are recirculated through the heat exchanger. 
Figure 8.44b shows, for comparison with Figure 8.44a, a forced circulation 
vacuum (flash cooling) crystallizer which operates without a heat exchanger, 
thus avoiding any problems with tube scaling. Crystallizers of this type are 
described in section 8.4.3. 

To minimize liquid droplet entrainment into the vapour space above the 
boiling liquid surface in a forced circulation crystallizer, a common cause of 
crystal encrustation on the walls, the evaporation rate must be controlled to 
limit the upward vapour velocity to below a critical value depending on the 
liquor and vapour densities, the latter quantity depending on the operating 
pressure in the vapour chamber (see section 9.3.1). Forced circulation crystal- 
lizers are widely used for a variety of substances such as NaCl, (NHz^SC^, 
Na2SC>4, FeSC>4, NiSC>4, citric acid, etc. In general, relatively small crystals 
(median size < 0.5 mm) are produced. 



Steam -c 





Figure 8.44. Forced circulation (a) evaporating crystallizer, (b) vacuum {flash cooling) 

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Industrial techniques and equipment 





Figure 8.45. Oslo-Krystal evaporator types: (a) waisted, (b) and (c) monolithic. A and B, 
circulating liquor outlet and inlet; C, vapour outlet; D, crystal magma outlet; E, liquor 
overflow. (After Bamforth, 1965) 

Oslo-Krystal evaporating crystallizer 

The principles of the Oslo-Krystal process, already referred to above in con- 
nection with cooling crystallizers, can also be applied to evaporative crystal- 
lization. Three forms of the Oslo-Krystal evaporating crystallizer are shown in 
Figure 8.45. The construction of these crystallizers, commonly used in multiple- 
effect systems, is of the 'closed' form, i.e. the vaporizer is directly connected 
with the crystallizer body to form a sealed unit. 

There are two basic types, the waisted (a) and the monolithic (b and c). One 
advantage of the latter is that the conical downcomer, which joints the vapor- 
izer and crystallizer sections and contains highly supersaturated (metastable) 
solution, is insulated from external conditions by mother liquor within the 
crystallizer. The circulating liquor inlet and outlet, A and B, are connected 
through a heat exchanger and pump. Liquor velocities of 1.5-2ms~' are 
commonly used through the tubes to minimize crystal depositions. 

Bamforth (1965) has given operating details of several Oslo-Krystal plants 
producing ammonium sulphate. Oslo-Krystal evaporating crystallizers have 
also been used for the manufacture of ammonium nitrate, borax, boric acid, 
sodium chloride, sodium dichromate and oxalic acid. In the case of sodium 
chloride, this type of crystallizer can, when operated under appropriate condi- 
tions, also produce spherical crystals of a median size 2-3 mm as opposed to the 
< 0.5 mm cubic crystals normally produced in a conventional forced circulation 
evaporating crystallizer. An example of this interesting and commercially useful 
crystalline form is shown in Figure 8.46 where a microscopic inspection of 

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384 Crystallization 


Figure 8.46. (a) A section through a 1 mm spherical crystal of sodium chloride showing an 
inner core of a 0.3 mm cubic crystal surrounded by a poly crystalline shell, (h) an SEM view 
of the surface of the outer shell. (Courtesy of ICI Ltd. Mond Division) 

a section through a 1mm sphere reveals at its centre a conventional 0.3 mm 
cubic crystal of salt, surrounded by a tightly bonded polycrystalline spherical 

Wetted-wall evaporative crystallizer 

A somewhat unusual application of the wetted-wall column, frequently used in 
gas-liquid mass transfer operations, has been reported by Chandler (1959). 
A hot concentrated solution is fed into a horizontal pipe, and cold air is blown 
in concurrently at a velocity of about 30ms~'. The liquid stream spreads over 
the internal surface of the pipe and cools, mainly by evaporation (see Figure 
8.47). The crystal slurry and air leave from the same end of the pipe. Only small 

6ft x 4in id gloss 
pipeline sections 


| Crystal slurry 
and air 


Figure 8.47. Arrangement of a wetted-wall evaporative crystallizer. (After Chandler, 1959) 


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Industrial techniques and equipment 


Table 8.2. Approximate theoretical capacity of a 4 in diameter wetted-wall 
evaporative crystallizer, based on a 3000 lbh -1 solution flow rate and a solution 
temperature drop from 100 to 67 °C (after J. L. Chandler, 1959) 


Crystal yield, lb h ' of anhydrous salt 


crystalline phase 

Due to cooling 

Due to 







CuS0 4 




5H 2 

CaCl 2 




2H 2 

CuCl 2 




2H 2 

A1 2 (S0 4 ) 3 




18H 2 

BaCl 2 




2H 2 

Ba(N0 3 ) 2 





K 2 S0 4 










CH 3 • COOK 




iH 2 

K 2 S0 4 • A1 2 (S0 4 ) 3 




24H 2 

MgS0 4 




6H 2 

MgCl 2 




6H 2 

MnCl 2 




2H 2 

(NH 4 ) 2 S0 4 










Na 3 P0 4 




12H 2 

Na 2 HP0 4 




2H 2 

NaH 2 P0 4 





CH 3 • COONa 





crystals can be produced by this method, and because of the evaporative loss of 
solvent only aqueous solutions can be handled. Nevertheless the equipment 
required is quite simple and relatively inexpensive. 

Although the pilot plant work described by Chandler was carried out on 
a small (100 cm diam.) unit and confined to the crystallization of sodium 
chloride, the wetted-wall crystallizer could be scaled up to larger sizes and used 
for other systems. The potential throughput of this small unit, however, could 
be quite high, depending on the temperature-solubility characteristics of the 
solute-solvent system, as indicated in Table 8.2. 

8.4.3 Vacuum crystallizers 

The term 'vacuum' crystallization is capable of being interpreted in many ways; 
any crystallizer that is operated under reduced pressure could be called a 
vacuum crystallizer. Some of the evaporators described above could be classi- 
fied in this manner, but these units are better, and more correctly, described 
as reduced-pressure evaporating crystallizers. The true vacuum crystallizer 
operates through flash cooling; supersaturation is achieved by simultaneous 
evaporation and cooling of the feed solution as it enters the vessel. 

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386 Crystallization 

To demonstrate the operating principles of these units, consider a hot satur- 
ated solution introduced into a lagged vessel maintained under vacuum. If the 
feed temperature is higher than that at which the solution would boil under the 
low pressure existing in the vessel, the feed solution will cool adiabatically to 
this temperature. The sensible heat liberated by the solution, together with any 
heat of crystallization liberated owing to the deposition of crystals at the lower 
temperature, causes the evaporation of a small amount of the solvent, which in 
turn results in the deposition of more crystals owing to the increased concen- 
tration. One of the attractions of 'vacuum' operation is the absence of a heat 
exchanger and the consequent elimination of tube-scaling problems. 

Vacuum crystallizers may be operated batchwise or continuously. In batch 
operation, the vessel is charged to a predetermined level with hot concentrated 
solution. As the pressure inside the vessel is reduced, the solution begins to boil 
and cool until the limit of the condensing equipment is reached. In order to 
increase the capacity of the condenser, and thereby increase the crystal yield, 
the vapour leaving the vessel can be compressed before condensation by the use 
of a steam-jet booster. The vacuum equipment usually consists of a two-stage 
steam ejector. 

Some form of agitation normally has to be provided to maintain reasonably 
uniform temperatures throughout the batch and to keep the crystals suspended 
in the liquor. Crystalline deposits around the upper portions of the inner walls 
of the vessel cause little inconvenience because, as the unit is operated batch- 
wise, the next charge will redissolve the deposit. When the batch reaches the 
required temperature, i.e. the desired degree of crystallization, it is discharged 
to a filtration unit. Small crystals, rarely much larger than about 250 um, are 
obtained from this type of crystallizer. 

In a continuously operated vacuum crystallizer the feed solution should 
reach the surface of the liquor in the vessel quickly, otherwise evaporation 
and cooling will not take place, because, owing to the hydrostatic head of 
solution, the boiling point elevation becomes appreciable at the low pressures 
(< 20 mbar) used in these vessels, and the feed solution will tend to migrate 
down towards the bottom outlet. Care must be taken, therefore, either to 
introduce the feed near the surface of the liquor in the vessel or to provide 
some form of agitation. 

As in the batch-operated units, crystalline deposits build up on the upper 
walls of the vessel. One way of overcoming this troublesome feature is to allow 
a small quantity of water to flow as a film down the walls at a rate less than the 
normal vaporization rate, so that no serious dilution of the charge occurs. 
A forced-circulation continuous vacuum crystallizer is shown in Figure 8.44b. 

Draft-tube agitation 

Vacuum crystallizers of the type shown in Fig. 8.44b generally produce small 
crystals (< 300 um) with a wide size distribution as a result of uncontrolled 
nucleation caused by a combination of feedstock 'flashing' on entry to the 
vessel and vigorous agitation. 


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Industrial techniques and equipment 


•» Vapour 


Figure 8.48. Swenson draft-tube-baffled (DTB) crystallizer 

A vacuum unit capable of producing larger crystals of narrow size distribu- 
tion is the Swenson draft-tube baffled (DTB) crystallizer (Figure 8.48) which has 
been described and compared with other types by Newman and Bennett (1959). 
A relatively slow-speed propeller agitator is located in a draft tube which 
extends to a few inches below the liquor level in the crystallizer. Hot, concen- 
trated feedstock enters at the base of the draft tube. The steady movement of 
magma and feedstock up to the surface of the liquor produces a gentle, uniform 
boiling action over the whole cross-sectional area of the crystallizer. The degree 
of supercooling thus produced is very low (< 1 °C), and in the absence of violent 
flashing, both excessive nucleation and salt build-up on the inner walls are 
minimized. The internal baffle in the crystallizer forms an annular space in 
which agitation effects are absent. This provides a settling zone that permits 
regulation of the magma density and control of the removal of excess nuclei. An 
integral elutriating leg may be installed underneath the crystallization zone (as 
depicted in Figure 8.48) to effect some degree of product classification. 

The DTB crystallizer can also be operated as a conventional evaporator- 
crystallizer by incorporating a steam-heated heat exchanger in the clear liquor 
recycle line. In this case, however, an elutriating leg is not normally installed 
and the recycled hot liquor stream enters directly into the draft-tube. Detailed 
design calculations for a forced circulation evaporative crystallizer of the DTB 
type to produce 75 ton/day of >200 urn urea crystals have been described by 
Bennett (1992). 


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388 Crystallization 

To vacuum 


Fresh solution 

Clear liquor 

Figure 8.49. Standard Messo turbulence crystallizer 

The Standard Messo turbulence crystallizer (Figure 8.49) is another draft- 
tube agitated unit. Two liquor flow circuits are created by concentric pipes: an 
outer ejector tube with a circumferential slot and an inner guide tube. Circula- 
tion is effected by a variable-speed agitator in the guide tube. The principle of 
the Oslo crystallizer is utilized in the growth zone; partial classification occurs 
in the lower regions, and fine crystals segregate in the upper region. The 
primary circuit is created by a fast upward flow of liquor in the guide tube 
and a downward flow in the annulus; liquor is thus drawn through the slot 
between the ejector tube and the baffle, and a secondary flow circuit is formed 
in the lower region of the vessel. Feedstock is introduced into the guide tube 
and passes into the vaporizer section where flash evaporation takes place. 
Nucleation, therefore, occurs in this region, and the nuclei are swept into the 
primary circuit. Mother liquor can be drawn off via a control valve, thus 
providing a means of controlling crystal slurry density. 

The Escher-Wyss Tsukishima double propeller (DP) crystallizer (Figure 8.50) 
is essentially a draft-tube agitated crystallizer with some novel features. The DP 
unit contains an annular baffled zone and a double-propeller agitator which 
maintains a steady upward flow inside the draft tube and a downward flow in 
the annular region. Very stable suspension characteristics are claimed. 

Fluidized-bed agitation 

As described above, the Oslo-Krystal unit is a fluidized-bed agitated crystal- 
lizer in which the gentle action minimizes secondary nucleation and allows 
large crystals to grow. Oslo-Krystal vacuum crystallizers can be of the 'open' 
(Figure 8.51) or 'closed' (Figure 8.45) types. In the former the crystallization 
zone is at atmospheric pressure. In the latter all parts of the equipment are 
under reduced pressure. 


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Industrial techniques and equipment 



NJ _^ Clear 
^" liquor 

Draft tube 

Elutriation leg 

Product crystals 

Figure 8.50. Escher-Wyss Tsukishima double propeller (DP) crystallizer 


Vaporiser — 


r— Vapour 

Solution manifold 

Fine salt 








Figure 8.51. An Oslo-Krystal vacuum crystallizer showing two different methods of 
operation: (a) classified suspension, (b) mixed suspension. (After Saeman, 1956) 

Two different methods of operation are shown in Figure 8.51 with (a) a 
classified suspension (circulating liquor) and (b) a mixed suspension (circulating 
magma). Classified operation, while capable of producing large regular 

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crystals, limits productivity because both the liquor velocity and the mass of 
crystals in suspension have to be restricted to keep the fines level below the 
pump inlet. Modification to magma circulation can improve the productivity 
considerably, because higher circulation rates and magma densities can be 
employed. Furthermore, the suspension volume is increased because magma 
circulates through the vaporizer and downcomer. In this type of operation, 
however, the bulk classifying action is lost, and it is necessary to provide 
a secondary elutriation zone in the suspension to permit segregation and 
removal of excess nuclei. Fines can be redissolved with live steam and the 
resulting solution fed to the vaporizer. 

One of the major factors in the successful operation of any controlled 
suspension crystallizer is the incorporation of a suitable fines trap. The earlier 
the excess nuclei are collected and destroyed the more efficient will be the 
process. In practice, fines are most economically removed when they reach 5 
to 10% of the average product size (Saeman, 1956). 

Multistage vacuum crystallizer 

The Standard Messo multistage vacuum crystallizer {Figure 8.52) provides 
a number of cooling stages in one vessel. The horizontal cylinder is divided 
into several compartments by vertical baffles that permit underflow of magma 
from one section to another but isolate the vapour spaces. Each vapour space is 
kept at its operating pressure by a thermocompressor, which discharges to 
a barometric condenser. 

Hot feedstock is sucked into the first compartment, which is operated at the 
highest pressure and temperature (say lOOmbar and 45 °C). Flash evaporation 
and cooling occur, and the resulting crystal slurry passes into the successive 
compartments, where the pressure is successively reduced and evaporation and 
cooling continue. In the last compartment the temperature and pressure may be 
10 °C and lOmbar, for example. Agitation is provided by the boiling action in 



Steam jet compressors « 




Air — 

I X I X X \ 
L— L-J—i * Crystal 

Figure 8.52. Standard Messo multistage vacuum crystallizer 

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Industrial techniques and equipment 391 

the compartments supplemented by air spargers. Mother liquor or magma is 
withdrawn from the last stage through a barometric leg or by means of a pump. 
Industrial units in a variety of sizes have been installed for the recovery of 
sodium sulphate from spin- bath liquors, the regeneration of pickling liquors, etc. 

8.4.4 Crystal yield 

The theoretical crystal yield for simple cooling or evaporating crystallization 
can be estimated from the solubility characteristics of the solution. For aqueous 
solutions, the following general equation applies (section 3.5): 

WR\c] - c 2 (l -V)] 
Y = ; 2K (8.23) 

1 - c 2 (R - 1) 

where c\ is the initial solution concentration, kg anhydrous salt per kg water; 
t'2 is the final solution concentration, kg anhydrous salt per kg water; W is the 
initial mass of water, kg; V is the water lost by evaporation, kg per kg of 
original water present; R is the ratio of molecular masses of hydrated to 
anhydrous salts; and Y is the crystal yield, kg. 

The actual yield may differ slightly from that calculated from equation 8.23. 
For example, if the crystals are washed with fresh solvent after filtration, losses 
may occur through dissolution. On the other hand, if mother liquor is retained 
by the crystals, an extra quantity of crystalline material will be deposited on 
drying. Furthermore, published solubility data usually refer to pure solvents 
and solutes. Because pure systems are rarely encountered industrially, solubil- 
ities should always be checked on the actual working liquors. 

Before equation 8.23 can be applied to vacuum (flash cooling) crystallization, 
the quantity V must be estimated: 

v = qR{c\ ~ c 2 ) + C(h - t 2 ){\ + d)[l - c 2 (R - 1)] 

A[l - c 2 (R - 1)] - qRc 2 l ' ) 

where A is the latent heat of evaporation of the solvent, Jkg~'; q is the heat of 
crystallization of the product, Jkg~'; t\ is the initial temperature of the solu- 
tion, °C; t 2 is the final temperature of the solution, °C; C is the specific heat 
capacity of the solution, Jkg~' K -1 ; and c\ and c 2 have the same meaning as in 
equation 8.23. 

As an example, the theoretical yield of sodium acetate crystals 
(CH3COONa-3H20) obtainable from a vacuum crystallizer operating with an 
internal pressure of 20 mbar, supplied with 2000 kg h~' of a sodium acetate 
solution (0.4 kg/kg of water) at 80 °C, may be calculated as follows. 

Boiling point elevation = 11.5 °C 

Heat of crystallization, q = 144kJ/kg of trihydrate 

Heat capacity of the solution = 3.5kJkg~' K~' 

Latent heat of water at 20 mbar, A= 2.46 M J kg -1 
Boiling point of water at 20 mbar = 17.5 °C 

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Solubilities of sodium acetate in water are given in the Appendix, Table A. 4. 

Equilibrium temperature of liquor = 17.5 + 11.5 = 29°C 

Initial concentration, c\ = 0.4/0.6 = 0.667 kg/kg of solvent 

Final concentration at 29 °C, C2 = 0.539 kg/kg of solvent 

Initial mass of water = 0.6 x 2000 = 1200 kg 

Ratio of molecular masses, R = 136/82 = 1.66 

The vaporization, V, is calculated from equation 8.24 

V = 0.1 53 kg/kg of water present originally 

This value of V substituted in equation 8.23 gives the crystal yield 

Y = 660 kg of sodium acetate trihydrate 

8.4.5 Controlled crystallization 

Selected seed crystals are sometimes added to a crystallizer to control the final 
product size. The effects of cooling rate and seeding are shown in Figure 8.53 
(Griffiths, 1925). When an unseeded solution is cooled rapidly (Figure 8.53a), 
cooling proceeds at constant concentration until the limit of the metastable 
zone is reached, where nucleation occurs. The temperature increases slightly 
due to the release of latent heat of crystallization, but cooling reduces it and 
more nucleation occurs. This results in a short period of instability after which 
the temperature and concentration subsequently fall as indicated. In such 
a process, both nucleation and growth are uncontrolled. 

Figure 8.53b demonstrates the slow cooling of a seeded solution in which 
temperature and solution composition are controlled within the metastable 
zone throughout the cooling cycle. Seeds are added as soon as the solution 
becomes supersaturated. Crystal growth occurs at a controlled rate only on the 
added seeds; spontaneous nucleation is avoided because the system is never 


Rapid cooling 
without seed/ 

Slow cooling 
with seed / 


Temperature - 



Figure 8.53. The effect of cooling rate and seeding on a crystallization operation: (a) 
uncontrolled and (b) controlled cooling 

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v Controlled 




Industrial techniques and equipment 393 

Controlled cooling 
. Natural cooling 


Figure 8.54. Cooling modes for a batch crystallizer. (a) natural cooling, (b) controlled 
(constant supersaturation) cooling 

allowed to become labile. This batch operating method is known as controlled 
crystallization; many modern large-scale crystallizers operate on this principle. 
If crystallization occurs only on the added seeds, the mass M s of seeds of size 
L s that can be added to a crystallizer depends on the required crystal yield Y 
(equation 8.23) and the product crystal size L p : 

M s 

YLlHL? - L]) 


The product crystal size from a batch crystallizer can also be controlled by 
adjusting cooling or evaporation rates. Natural cooling (Figure 8.54a), for 
example, produces a supersaturation peak in the early stages of the process 
when rapid, uncontrolled heavy nucleation inevitably occurs. However, nucle- 
ation can be controlled within acceptable limits by following a cooling path 
that maintains a constant low level of supersaturation (Figure 8.54b). 

The calculation of optimum cooling curves for different operating conditions 
is complex (Mullin and Nyvlt, 1971; Jones and Mullin, 1974), but the following 
simplified relationship is often adequate for general application: 

9o ~ (Oo - t )(t/T) 3 


where 6q, Of, and 6, are the temperatures at the beginning, end, and any time t 
during the process, respectively, and r is the overall batch time. 

The subject of controlled (programmed) crystallization is considered in more 
detail in sections 9.1.4 and 9.9 (example 9.3). 

8.4.6 Miscellaneous crystallization techniques 

Salting-out crystallization 

A solute can be deposited from solution by the addition of another substance 
(a soluble solid, liquid or gas) which effectively reduces the original solute 

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394 Crystallization 

solubility. The process is often referred to as 'salting out', although it applies to 
electrolytes and non-electrolytes alike. A slow addition of the salting-out agent 
can change a fast precipitation of the solute into a more controlled crystal- 
lization process. For convenience, this topic is dealt with in more detail as one 
of the techniques of precipitation in section 8.1.5. 

Several potential applications of salting-out crystallization have been 
reported for the production of pure inorganic salts from aqueous solution 
(Gee, Cunningham and Heindl, 1947), the recovery of fertilizer-grade salts from 
seawater (Fernandez-Lozano, 1976) and the separation of inorganic salt mix- 
tures (Alfassi and Mosseri, 1984). The recovery of inorganic salts from concen- 
trated aqueous solution is considered by Weingaertner, Lynn and Hanson 
(1991) who propose a process in which the filtered mother liquor is regenerated 
into two phases, aqueous and organic, by change of temperature and addition 
of more feedstock. Both phases are then recycled. Detailed examples are 
given for the recovery of sodium chloride using 2-propanol or diisopropylamine 
and sodium carbonate using 1-propanol or 1-butanol. The kinetics of the salt- 
ing-out crystallization of anhydrous sodium sulphate, from dilute aqueous 
solutions of sulphuric acid using methanol, have been studied by Mina- 
Mankarios and Pinder (1991). The effects of trace quantities of Cr 3+ were also 

Reaction crystallization 

The production of a solid crystalline product as the result of chemical reaction 
between gases and or liquids is a standard method for the preparation of many 
industrial chemicals. Crystallization occurs because the gaseous or liquid phase 
becomes supersaturated with respect to the reaction product. A precipitation 
operation (section 8.1.5) can be transformed into a crystallization process by 
moderation and control of the degree of supersaturation. 

Reaction crystallization is practised widely, especially in industries where 
valuable waste gases are produced. For instance, ammonia can be recovered 
from coke oven gases by converting it into ammonium sulphate by reaction 
with sulphuric acid. Agitation of the crystals within the reaction vessel is 
effected by a combination of the vigorous nature of the exothermic reaction 
and air sparging. The heat of reaction is removed by the evaporation of water 
added to the reaction zone. Bamforth (1965) gives a detailed heat and materials 
balance for a commercial installation producing 32tonh~' of (NH^SCV 

Another example of a reaction crystallizer is the carbonation tower method 
used for the production of sodium bicarbonate by the interaction between brine 
and flue gases containing about 10-20 per cent of carbon dioxide. A 15 m high 
tower is kept full of brine, and the flue gas, which enters at the bottom of the 
tower, flows upwards countercurrently to the brine flow. Carbonated brine is 
pumped continuously out of the bottom of the tower. To effect efficient 
absorption of CO2, internal rotating screens continually redisperse the gas 
stream in the form of tiny bubbles in the liquor. The operating temperature is 
controlled at about 38 °C, which has been found from experience to give both 
good absorption and crystal growth (Hou, 1942; Garrett, 1958). 

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Industrial techniques and equipment 395 

A more recent example is flue-gas desulphurization, an environmental pro- 
tection process employed for the removal of SO2 from coal-fired power station 
flue gases. One widely used method is to absorb the SO2 in an aqueous 
suspension of finely crushed limestone in an agitated vessel or spray tower. 
The resulting CaSCh solution is passed to an air-sparged tank where it is 
oxidized to precipitate CaS04 • 2H2O which can find use in cement or wall- 
board manufacture. Another desulphurization technique can produce a fertil- 
izer-grade (NH4) 2 S04 by countercurrent scrubbing flue gases with aqueous 
ammonia (Wallach, 1997; AIChE, 2000). 

A number of laboratory studies have been recorded recently aimed at char- 
acterizing the kinetics of both the chemical reaction and crystallization steps in 
a reaction crystallization process. Examples of liquid phase reactions studied 
for this purpose are the crystallization of salicylic acid from aqueous solutions 
of sodium salicylate using dilute sulphuric acid (Franck et al., 1988) and the 
crystallization of various calcium phosphates by reacting equimolar aqueous 
solutions of calcium nitrate and potassium phosphate (Tsuge, Yoshizawa and 
Tsuzuki, 1996). Several aspects of crystal size distribution control in semi-batch 
reaction crystallization have been considered by Aslund and Rasmuson (1990) 
who studied the crystallization of benzoic acid by reacting aqueous solutions of 
sodium benzoate with HC1. An example of crystallization arising from a gas- 
liquid reaction in an aqueous medium is the precipitation of calcium carbonate 
from the reaction between calcium hydroxide and CO2 (Wachi and Jones, 

Adductive crystallization 

The simple crystallization of a binary eutectic system only produces one of the 
components in pure form, while the residual mother liquor composition pro- 
gresses towards that of the eutectic (section 4.3.1). There is often a need, 
however, to produce both components in pure form, and one way in which 
this may be achieved is to add a third component to the system which forms a 
compound with one of the binary components. Phase diagrams for systems 
with compound formation are discussed in section 4.3.2. 

A typical sequence of operation would be as follows. A certain substance X is 
added to a given binary mixture of components A and B so that a solid 
complex, say A • X, is precipitated. Component B is left in solution. The solid 
and liquid phases are separated, and the solid complex is split into pure A and 
X by the application of heat or by dissolution in some suitable solvent. 

The best-known example of compound formation in solvent-solute systems 
is the formation of hydrates, but other solvates, e.g. with methanol, ethanol and 
acetic acid, are known. In these cases the ratio of the molecules of the two 
components in the solid solvate can usually be expressed in terms of small 
integers, e.g. CUSO4 • 5H2O or (CeHs^NH • (CeHs^CO (diphenylamine • ben- 
zophenone). There are, however, several other types of molecular complex that 
can be formed which are not necessarily expressed in terms of simple ratios. 
These complexes are best considered not as chemical compounds but as 
strongly bound physical mixtures. 

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396 Crystallization 

Clathrate compounds are of this type: molecules of one substance are 
trapped in the open structure of molecules of another. Hydroquinone forms 
clathrate compounds with SO2 and methanol, for example. Urea and thiourea 
also have the property of forming complexes, known as adducts, with certain 
types of hydrocarbons. In these cases molecules of the hydrocarbons fit into 
'holes' or 'channels' in the crystals of urea or thiourea; the shape and size of the 
molecules determine whether they will be adducted or not. 

It is open to question whether adduct formation can really be considered as 
a true crystallization process, but the methods of operation employed are often 
indistinguishable from conventional crystallization methods. Several different 
names have been given to separation techniques based on the formation of 
adducts, but the name adductive crystallization is probably the best and will be 
used here. 

Several possible commercial applications of adductive crystallization have 
been reported. For example, the system m-cresol-/>-cresol forms two eutectics 
over the complete range of composition, and the separation of the pure com- 
ponents cannot be made by conventional crystallization. Savitt and Othmer 
(1952) described the separation of mixtures of these two components by the use 
of benzidine to form a solid addition compound with />-cresol. Actually both 
m-cresol and />-cresol form addition compounds with benzidine, but the meta- 
compound melts at a lower temperature than the para-. If the process is carried 
out at an elevated temperature the formation of the meta- complex can be 
avoided. The method consists of adding benzidine to the meta-para- mixture at 
1 10 °C. The/>-cresol ■ benzidine crystallizes out, leaving the m-cresol in solution. 
The crystals are filtered off, washed with benzene to remove w-cresol, and the 
washed cake is distilled under reduced pressure to yield a 98 per cent pure 

In a somewhat similar manner /(-xylene can be separated from a mixture of 
m- and /^-xylene; this binary system forms a eutectic. Carbon tetrachloride 
produces an equimolecular solid compound with />-xylene, but not with o- or 
w-xylene. Egan and Luthy (1955) reported on a plant for the production of pure 
/(-xylene by crystallization meta-para- xylene mixtures in the presence of carbon 
tetra-chloride. Up to 90 per cent of the para- isomer was recovered by distilla- 
tion after splitting the separated solid complex. The meta- isomer was recovered 
by fractionally crystallizing the CCU-free mother liquor. Perfect separation 
of />-xylene is not possible, because the ternary system CCU/m-xylene/CCU- 
/^-xylene forms a eutectic, but fortunately the concentration of the complex 
CCI4 • /^-xylene in this eutectic is very low. Several commercial clathration 
processes for the separation of w-xylene from Cs petroleum reformate fractions 
using a variety of complexing agents have been operated (Sherwood, 1965). 

Separation processes based on the formation of adducts with urea, thiourea 
and other substances have been described by several authors (Findlay and 
Weedman, 1958; Hoppe, 1964; Santhanam, 1966; McCandless, 1988; Jadhav, 
Chivate and Tavare, 1991, 1992; Kitamura and Tanaka, 1994). 

Urea forms addition complexes with straight-chain, or nearly straight-chain, 
organic compounds such as paraffin and unsaturated hydrocarbons (> 6 
carbon atoms), acids, esters and ketones. Thiourea forms rather less stable 

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Industrial techniques and equipment 397 

complexes with branched -chain hydrocarbons and naphthenes, e.g. cyclohex- 
ane. For example, if a saturated aqueous solution of urea in methanol is added 
to an agitated mixture of cetane and iso-octane, a solid complex of cetane-urea 
is formed almost immediately, deposited from the solution in the form of fine 
needle crystals. The iso-octane is left in solution. After filtration and washing 
the complex is heated or dissolved in water, and pure cetane is recovered by 
distillation. If thiourea is used instead of urea, iso-octane can be recovered 
leaving the cetane in solution. 

Extractive crystallization 

As described above, adductive crystallization is one method for separating 
a binary eutectic mixture into its component parts. Another possibility is to 
alter the solid-liquid phase relationships by introducing a third component, 
usually a liquid called the solvent. This process is known by the name 'extract- 
ive crystallization'. 

Chivate and Shah (1956) discussed the use of extractive crystallization for the 
separation of mixtures of m- and />-cresol, a system in which two eutectics are 
formed. Acetic acid was used as the extraction solvent. Details of the relevant 
phase equilibria encountered in the various combinations of systems were given 
together with an account of the laboratory investigations. While it was shown 
that acetic acid is not a particularly good solvent for the separation process in 
question, it was clearly indicated that extractive crystallization, provided 
a suitable solvent is chosen, has a large number of potential applications. 
Dikshit and Chivate (1970) have reported ternary phase equilibria for the separation 
of o- and /?-nitrochlorobenzenes by extractive crystallization with /?-dichlor- 
benzene, and have proposed general methods for predicting the selectivity of 
a solvent. 

A procedure for screening solvents for both adductive and extractive crystal- 
lization has been proposed by Tare and Chivate (1976a) who also described 
(1976b) adductive and extractive routes towards the separation of o- and 
/>-nitrochlorobenzene using />-dibromobenzene. Gaikar, Mahapatra and 
Sharma (1989) have described procedures for the separation of other close- 
boiling organic mixtures. Rajagopal, Ng and Douglas (1991) have formulated 
a systematic procedure for extractive crystallization separations. Design equa- 
tions are proposed for flow-sheet analysis and design variables and constraints 
are identified. A worked example is outlined for the recovery of /^-xylene from 
a m-xylene mixture using pentane as the solvent. In a further contribution, 
Dye and Ng (1995) demonstrate that extractive crystallization can offer a way 
to by-pass eutectics in multicomponent systems and to recover individual 

Weingaertner, Lynn and Hanson (1991) have assessed the recovery of inor- 
ganic salts from concentrated aqueous solution in a process that they called 
'extractive crystallization', but in part follows conventional salting-out pro- 
cedures (sections 8.1.5 and 8.4.6). Enhanced recovery of salt and economical 
use of solvent are achieved, however, by generating a two-phase change in the 
filtered mother liquor, by temperature and composition manipulation, and the 

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398 Crystallization 

use of recycling. Amongst the examples examined are the recovery of sodium 
chloride using 2-propanol and sodium carbonate using 1-propanol. 

Freeze crystallization 

Large-scale crystallization by freezing has been practised commercially since the 
1950s when the first successful continuous column crystallizers were developed 
for use in the petrochemicals industry, particularly for />-xylene. Some of these 
processes have already been discussed in section 8.2.2. The present section will 
be devoted to the freezing of aqueous systems and the removal of water, as ice, 
either as the required product or as the unwanted component of the mixture. 

One of the great engineering challenges of the present day is the search for 
the ideal process to produce fresh water from seawater. Desalination by crystal- 
lization offers several possible routes. The main freezing processes being pur- 
sued at the present time (Barduhn, 1975) are 

1 . Vacuum flash freezing 

2. Hydrate freezing 

3. Immiscible refrigerant freezing 

The first of these processes utilizes the cooling effect of vaporization. Seawater 
is sprayed into a low-pressure chamber, where some of the water vapour flashes 
off and the brine partially freezes. The ice-brine slurry is separated, and the ice 
crystals are washed and remelted. Commercial units based on this principle 
have been operated. 

The hydrate processes utilize the fact that certain substances can form inclu- 
sion compounds with water, loosely called 'hydrates', as described above in the 
section dealing with adductive crystallization. These crystalline substances are 
separated from the residual brine and decomposed to recover the hydrating agent. 
Propane is one of the most promising hydrating agents. One of the important 
advantages of the hydrate processes is that they operate close to ambient 
temperature (10-15 °C) and energy costs are minimized. One of the main dis- 
advantages is the difficulty in growing crystals with good filtration and washing 
characteristics. Light, feathery crystals are quite common, and the compacted 
beds have a low permeability. The success of the hydrate processes, therefore, 
appears to hinge on developing a reliable method of controlling the crystal habit. 

The indirect freezing processes, i.e. those in which brine is crystallized in 
some form of heat exchanger, have largely been abandoned, but the direct 
contact between brine and refrigerant is a very promising technique. The direct 
contact between liquid «-butane (which does not form a 'hydrate') and seawater 
has been widely investigated (Denton et ah, 1970). One approach is shown in 
Figure 8.55. Precooled brine (3.5 per cent dissolved salts) is fed to the crystal- 
lizer and liquid butane, which is sparged into the slurry, boils under reduced 
pressure (0.86 bar). Agitation is not usually necessary in the crystallizer in view 
of the vigorous boiling action of the butane. The slurry of ice crystals is pumped 
to the wash column. Butane vapour leaving the crystallizer is compressed (1.1 
bar) before entering the ice melter where it is liquefied and, after separation 
from the water, is pumped back into the crystallizer. The compressor, therefore, 


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Industrial techniques and equipment 


-— Reject brine T.D.S. 7% 
— Product 



100 ppm 

.-. compressor llDor jS^Tf" 

Debutanisers f S~~ — "- ""!T ;S~ ^ 



feed 1 
3.5% x 

25 ' C h»jt|| 


wa ter nvie lter 


12.8 bar 


Product water 

Liquid butane 

Figure 8.55. Immiscible refrigerant (n-butane) freeze desalination process (U.K. A.E.R.E., 

provides the energy input to the system. A hydraulic piston wash column 
provides a simple method for both separating the bulk of the mother liquor 
from the ice and providing countercurrent flow for washing the ice crystals free 
from residual brine. 

An attraction of freezing over evaporation for the removal of water from 
solutions is the potential for saving heat energy: the enthalpy of crystallization 
of ice (334 k J kg -1 ) is only one seventh of the enthalpy of vaporization of water 
(2260 kJkg~'). In practice, however, the overall process energy consumptions, 
taking into account the other ancillary separation operations needed, are often 

A definite advantage of freeze crystallization, important in many food 
industry applications, is that volatile flavour components that are normally 
lost during conventional evaporation can be retained in a freeze-concentrated 
product. In fact, at present, freeze crystallization finds its main application in 
the food industry, for the concentration of fruit juices, etc. Indirect-contact 
freezing processes are normally used, e.g. the liquid feedstock is crystallized in a 
scraped-surface heat exchanger (section 8.2.2) and the resulting ice slurry passes 
to a wash column where the crystals are separated and washed to recover 
valuable product. The wash column is the key item in the process. Figure 8.56 
shows an example of the Grenco system of freeze crystallization. 

Reviews of the potential of industrial freeze crystallization have been made 
by Heist (1981) and Casper (1981). 

The subject of gas hydrates has become highly topical in recent years (Sloan, 
1997), particularly since the discovery of vast amounts of natural gas hydrates 
under ocean floor sediments at depths >500 m and in polar permafrost regions. 
Gas hydrates are clathrate compounds in which variable (non-stoichiometric) 
amounts of gas, e.g., methane, ethane and propane, are trapped within ice 
crystal lattice 'cages'. The amount of entrapped gas increases with lowering 
temperature and increasing pressure. It has been estimated that world-wide the 
amount of methane trapped in gas hydrates is around 2 x 10 16 m 3 at STP, 
which is roughly equivalent to twice the mass of carbon in all conventional 

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product out 

water out 


1o be 

Figure 8.56. Grenco freeze concentration process (single stage). 1, scraped surface heat 
exchanger; 2, agitated r eery stallizer; 3, wash column with bottom perforated piston and top 
ice scraper; 4, circulating pumps; 5, concentrated product expansion vessel; 6, ice melter 

gas, oil and coal deposits combined. Although the commercial recovery of 
methane from natural hydrate sources has not yet been achieved, very active 
international research programmes are at present being pursued (Max and 
Dillon, 2000). 

Emulsion crystallization 

Organic substances may be purified by fractional crystallization from the melt 
or from organic solvents, but these operations frequently present difficulties in 
large-scale production. In the latter case, apart from expensive solvent losses 
and the potential fire and explosion hazards, yields are often low on account of 
the high solubility of the crystals. A method known as emulsion crystallization 
described by Holeci (1965) appears to be free from these shortcomings. 

Briefly, crystallization is carried out by cooling from an aqueous emulsion. 
Impurities, in the form of eutectic mixtures, remain in the emulsion, from which 
they may be recovered by further cooling. The organic substance should be 
(a) practically insoluble in water and (b) able to melt and solidify in a hetero- 
geneous aqueous medium, and remain stable. 

The organic melt is emulsified in water with the aid of a suitable nonionic 
agent and stabilized by a protective colloid, e.g. potato starch gelatinized with 
water. The system is crystallized by cooling, and the crystals are separated from 
the emulsion and washed with water. The operation may be repeated if 
required. A typical example is shown in Figure 8.57 where five or six emulsion- 
crystallization cycles yield an almost pure naphthalene at an overall yield of 
70 per cent compared with a less pure product at a 2 per cent yield by seven 
conventional fractional crystallization steps. The high efficiency of the emul- 
sion crystallization is apparently due to the fact that crystal agglomeration does 

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Industrial techniques and equipment 















x Fractional 
o Emulsion 


I 2 3 4 5 6 7 8 

Number of crystallizations 

Figure 8.57. Comparison of fractional and emulsion crystallization cycles for the 
purification of naphthalene . (After Holeci, 1965) 

not occur to any great extent and the impure emulsion is readily washed away 
with water. 

Although there has not been any published report of commercial exploitation 
of this technique so far, other reports along similar lines tend to suggest that 
there could still be potential applications for emulsion-phase crystallization 
procedures (Sun, 1971; Dow Chemical Co., 1978; Davies, Ming and Garside, 
1990; Davey, Hilton and Garside, 1997; Espitalier et al, 1997). 

Spray crystallization 

The term 'spray' crystallization is really a misnomer; strictly speaking, indi- 
vidual crystals are not grown by this method - solid is simply deposited from 
a very concentrated solution by a technique similar to that used in spray drying. 
Size and shape of the solid particles depend to a large extent on those of the 
spray droplets. The spray method is often employed when difficulties are 
encountered in the conventional crystallization techniques, or if a product with 
better storage and handling properties can be produced. 

Anhydrous sodium sulphate can be manufactured by spray crystallization. 
Below 32.4 °C sodium sulphate crystallizes from solution in the form of the 
decahydrate: above this temperature the anhydrous salt is formed. However, 
anhydrous sodium sulphate has an inverted temperature-solubility character- 
istic (see Figure 3.1b), and trouble is encountered with scale formation on the 
heat transfer surfaces of conventional evaporating crystallizers when operating 
temperatures in excess of 32.4 °C are used. In a plant described by Holland 
(1951) a concentrated solution of sodium sulphate was sprayed, or splashed, 
in the form of tiny droplets into a chamber through which hot gases flowed. 
The gases entered at about 900 °C. The continuously operated unit produced 
a powdered anhydrous product. 

In recent years an increasing interest has been paid to the utility of the spray 
technique for commercial production purposes, and a number of studies have 
been made to ascertain the important variables affecting the growth of solid 
particles in gas-solid fluidized beds. Metheny and Vance (1962), for example, 

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402 Crystallization 

have reported pilot-scale trials for the production of crystalline granules of 
(NH4) 2 S04, NaCN, CaCl2 and NaOH by the spraying of solution onto seed 
crystals introduced into a fluidized bed. Similar studies have been made with 
Na 2 S0 4 • 10H 2 O, A1(N0 3 ) 3 , CaCl 2 and Ca(N0 3 ) 2 ■ 4H 2 (Markvart, Vanecek 
and Drbohlav, 1962). 

Analyses of the kinetics of particle growth in fluidized beds have been made 
by several authors (Grimmett, 1964; Shakova et ah, 1973; Uemaki and Mathur, 
1976; Smith and Nienow, 1983). 

The spray crystallization of a solution has some features in common with 
prilling, the melt crystallization technique described in section 8.2.2. 

Spherical crystallization 

Fine crystals of pharmaceutical substances are frequently granulated to impart 
good flow properties and to assist tableting procedures. An interesting tech- 
nique for transforming fine crystals into dense spherical agglomerates during 
the crystallization process, hence the name spherical crystallization, appears to 
have potential application in the pharmaceutical industry. 

The method is based on the addition to a crystallizing system of a small 
amount of an immiscible liquid that preferentially wets the developing fine 
crystals and encourages them to compact into spherical agglomerates ~ 250- 
1000 (im. The technique stems from work that established successful granula- 
tion procedures, starting with dry crystals, using carbon tetrachloride as 
a 'bridging agent' (Capes and Sutherland, 1967), but chloroform appears to 
be the preferred organic liquid for use in association with crystallization from 
aqueous solution. An example of spherical crystallization is that of salicylic 
acid which may be agglomerated by adding an ethanolic solution of it to 
a well-dispersed 30 : 1 mixture of water and chloroform. Alternatively an aque- 
ous solution of salicylic acid may be added to a solution of chloroform in 
ethanol. Several pharmaceutical substances have been successfully processed 
by this technique (Kawashima, 1984; Kawashima, Okumura and Tekenaka, 

Kerosene has been used as a binding agent for the production of spherical 
agglomerates of precipitated inorganic substances, e.g. calcium carbonate (Bos 
and Zuiderweg, 1985; Kawashima et al., 1986). 

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9 Crystallizer design and operation 

Many of the difficulties facing designers of industrial crystallizers arise from the 
shortage of basic data in the technical literature. However, not only are pub- 
lished data scarce, they are so frequently unreliable. It is not uncommon to find 
different investigators reporting crystal growth rates for the same substance 
differing by an order of magnitude or more. In such cases it is often impossible 
to select the appropriate value for a given situation, usually because some 
important parameter has not been specified, or perhaps not even measured. 
Reliable nucleation data applicable to industrial systems are seldom, if ever, 
available in the literature. 

The unit operation of crystallization is governed by some very complex 
interacting variables. It is a simultaneous heat and mass transfer process 
with a strong dependence on fluid and particle mechanics. It takes place in a 
multiphase, multicomponent system. It is concerned with particulate solids 
whose size and size distribution, both incapable of unique definition, vary with 
time. The solids are suspended in a solution which can fluctuate between a so- 
called metastable equilibrium and a labile state, and the solution composition 
can also vary with time. The nucleation and growth kinetics, the governing 
processes in this operation, can often be profoundly influenced by mere traces 
of impurity in the system; a few parts per million may alter the crystalline 
product beyond all recognition. 

It is perhaps no wonder that crystallization was once called an art rather 
than a science, but to do so today would be to ignore the tremendous 
advances that have been made over the past three or more decades in both 
the theory and practice of the subject. The object of this chapter is to 
describe some of the recent developments in the area of crystallizer design 
and operation and to demonstrate how simple laboratory scale tests, some of 
which have been described in earlier chapters, can provide basic design 

9.1 Crystal size distribution (CSD) 

One of the earliest investigations aimed at studying the size distribution of 
crystals in a continuous crystallizer was made by Montillon and Badger (1927) 
with Na 2 S0 4 • 10H 2 O and MgS0 4 • 7H 2 0. Shortly afterwards McCabe (1929) 
analysed the problem of crystal size distribution (CSD) and developed the AL 
law, making the following assumptions: (a) all crystals have the same shape; 
(b) they grow invariantly, i.e. the growth rate is independent of crystal 
size; (c) supersaturation is constant throughout the crystallizer; (d) no nucle- 
ation occurs; (e) no size classification occurs in the crystallizer; (f ) the relative 
velocity between crystals and liquor remains constant. For a full account of the 

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404 Crystallization 

derivation of the AL law reference should be made to the original paper, but 
the main steps are briefly as follows. 

The mass of one crystal of a chosen characteristic size L (section 2.14.2) is 
given by ap c L 3 , where a is a volume shape factor (section 2.14.3) and p c is the 
crystal density. The number of crystals, dN, of size L in a mass dM is thus 
dN = dM jap c L? . Assuming no nucleation, the number of seeds dN s of size L s 
is equal to the number of product crystals dN p of size L p , i.e. 

dM s dM p dA/p 


ap c L 3 ap c L 3 ap c (L s + AL) 3 
where AL is the growth increment. Therefore 


L s + AL\ 3 

d.l/, = |-^ JdM t 

-M, / a , \3 

where M p is the product crystal yield obtained from an initial mass of seed 
crystals, M s . 

McCabe recognized that the ideal conditions assumed in this derivation are 
unlikely to be attained in a real crystallizer. Temperature and supersaturation 
gradients are unavoidable; invariant growth is comparatively rare; different 
crystal faces usually grow at